Math Problems Vault

Problem 1.10.1

Let the function $f$ be defined in a neighborhood of the point $x_{0}$ . Its increment $\Delta f$ is an infinitesimal function of the increment of the argument $\Delta x$ as $\Delta x \rightarrow 0$ . Is it true that the function $f$ is continuous at the point $x_{0}$ ? Is the converse true?

?

Problem 1.10.2

Let the function $f$ be defined in a neighborhood of the point $x_{0}$ . Formulate, in " $\varepsilon -\delta$ language," the negation of the statement that: а) $\lim_{x \rightarrow x_{0}} f(x) = f\left(x_{0}\right)$ ; б) $\lim_{\Delta x \rightarrow 0} \Delta f = 0$ .

?

Problem 1.10.3

а) Let the function $f$ be defined on the interval $\left(a ; x_{0}\right]$ . Formulate, in " $\varepsilon - \delta$ language," the negation of the statement that $f$ is continuous from the left at the point $x_{0}$ . б) Let the function $f$ be defined on the interval $\left[x_{0} ; b\right)$ . Formulate, in " $\varepsilon -\delta$ language," the negation of the statement that $f$ is continuous from the right at the point $x_{0}$ .

?

Problem 1.10.4

Let the function $f$ be defined in a neighborhood of the point $x_{0}$ . Can a function be continuous at the point $x_{0}$ from the right and not continuous there from the left?

?

Problem 1.10.5

Prove that the function $y(x)$ is continuous at every point of its domain, if:

?

(1)

$y = 2 x-1$ ;

(2)

$y = x^{2}$ ;

(3)

$y = \sqrt{x}$ ;

(4)

$y = 1 / x$ ;

(5)

$y = a x+b, a \neq 0$ ;

(6)

$y = \left|x\right|$ ;

(7)

$y = x^{3}$ ;

(8)

$y = \sqrt[3]{x}$ ;

(9)

$y = 1 / x^{2}$ .

Problem 1.10.6

(1)(2)(3)

Using the inequality $\left|\sin x\right| \leqslant \left|x\right|$ , prove the continuity of the function $y$ at every point of its domain, if:

?

(1)

$y = \sin x$ ;

(2)

$y = \cos x$ ;

(3)

$y = \sin (a x+b)$ .

Problem 1.10.7

(1)(2)

Formulate, in " $\varepsilon -\delta$ language," the definition of continuity:

?

(1)

from the left;

(2)

from the right, at the point $x_{0}$ , of a function; and write it using the symbols $\exists , \forall$ .

Problem 1.10.8

(1)(2)

Prove that:

?

(1)

the function $\sqrt[4]{1-x}$ is continuous at every point $x_{0} < 1$ and continuous from the left at the point $x_{0} = 1$ ;

(2)

the function $x \mathbb {E}\left[x\right]$ is continuous from the right at every point $x_{0} = n$ , $n \in \mathbb {Z}$ .

Problem 1.10.9

Using the properties of limits and the results of problems 5, 6, prove the continuity of the function $y(x)$ at every point of its domain, if:

?

(1)

$y = \frac{2 x-1}{x^{2}+2}$

(2)

$y = \operatorname {tg} x$

(3)

$y = x^{2}+2 \sin x$

(4)

$y = \sqrt{x} \sin 2 x$ ;

(5)

$y = x^{n}, n \in \mathbb {N}$ ;

(6)

$y = 1 / x^{n}, n \in \mathbb {N}$ .

Problem 1.10.10

Using the monotonicity of the function $y = a^{x}, x \in \mathbb {R}$ , and the fact that $\lim_{n \rightarrow \infty } a^{1 / n} = 1$ (see §8, Example 9 and problem 60), prove the continuity, at every point $x \in \mathbb {R}$ , of the function $y = a^{x}, x \in \mathbb {R}$ .

?

Problem 1.10.11

The function $f$ is defined in a neighborhood of the point $x_{0}$ , except at the point $x_{0}$ itself. Extend the function $f$ by assigning a value to $f\left(x_{0}\right)$ so that the resulting function is continuous at the point $x_{0}$ , if:

?

(1)

$f(x) = \frac{x^{2}-1}{x+1}, x_{0} = -1$ ;

(2)

$f(x) = \frac{x^{3}-1}{x^{2}-1}, x_{0} = 1$

(3)

$f(x) = \frac{\sqrt{1+x}-1}{x}, x_{0} = 0$ ;

(4)

$f(x) = \frac{\sin x}{x}, x_{0} = 0$ ;

(5)

$f(x) = x \operatorname {ctg} x, x_{0} = 0$ ;

(6)

$y = \frac{1-\cos x}{x^{2}}, x_{0} = 0$ .

Problem 1.10.12

Formulate the definition of a function continuous at a point, using the Heine definition of the limit of a function (i.e., in terms of sequences).

?

Problem 1.10.13

Prove that if a function is continuous at a point, then it is bounded in some neighborhood of this point.

?

Problem 1.10.14

Let the function $f$ be continuous at the point $x_{0}$ and $f\left(x_{0}\right) \neq 0$ . Prove that there exist a number $C > 0$ and a neighborhood of the point $x_{0}$ such that for any $x$ in this neighborhood the inequality $\left|f(x)\right| \geqslant C$ holds.

?

Problem 1.10.15

The function $f$ is continuous at the point $x_{0}$ , and in every neighborhood of this point there are both values of $x$ at which the function is positive and values of $x$ at which the function is negative. Find $f\left(x_{0}\right)$ .

?

Problem 1.10.16

Let a function be defined in a neighborhood of the point $x_{0}$ . Formulate, using the symbols $\exists , \forall$ , the statement that the function is not continuous at the point $x_{0}$ .

?

Problem 1.10.17

Prove that the function $f$ is not continuous at the point $x_{0}$ ; plot the graph of this function, if:

?

(1)

$f(x) = \begin{cases} x+1, & x > 0, \\ x^{2}, & x \leqslant 0,\end{cases} \quad x_{0} = 0 ;$

(2)

$f(x) = \begin{cases} 1 / x, & x \neq 0, \\ 0, & x = 0,\end{cases} \quad x_{0} = 0$

(3)

$f(x) = \operatorname {sign}(x+1), x_{0} = -1$ ;

(4)

$f(x) = \begin{cases} 1 / x^{2}, & x \neq 0, \\ 1, & x = 0,\end{cases} \quad x_{0} = 0 ;$

(5)

$f(x) = \mathbb {E}\left[x\right], x_{0} = 2$ .

Problem 1.10.18

Find the points of discontinuity of the function, determine their type, find the jumps of the function at the discontinuities of the first kind, and plot the graph of the function:

?

(1)

$f(x) = \begin{cases} 1/(x-1), & x < 0, \\ (x+1)^{2}, & 0 \leqslant x \leqslant 2, \\ 1-x, & 2 < x; \end{cases}$

(2)

$y = \dfrac {\left|x+2\right|}{x+2}$ ;

(3)

$y = \frac{1}{x^{2}-4}$ ;

(4)

$y = \frac{\left|x-1\right|}{x^{2}-x^{3}}$ ;

(5)

$y = x-\mathbb {E}\left[x\right]$ ;

(6)

$y = \frac{1}{x-\mathbb {E}\left[x\right]}$ ;

(7)

$y = \frac{(x+1)^{2}-(x-1)^{2}}{x^{2}-x}$ ;

(8)

$y = \frac{2 \operatorname {sign}(1-x)}{(\operatorname {sign}(x+1))^{2}(x+1+(x-1) \operatorname {sign} x)}$ ;

(9)

$y = \frac{1}{\cos x}$ ;

(10)

$y = \frac{x}{\sin x}$ .

Problem 1.10.19

(1)(2)(3)(4)

Determine whether a value of $a$ exists for which the function $f$ is continuous at the point $x_{0}$ , if:

?

(1)

$f(x) = \begin{cases} x \sin (1 / x), & x \neq 0, \\ a, & x = 0, \quad x_{0} = 0 ;\end{cases}$

(2)

$f(x) = \begin{cases} \frac{1+x}{1+x^{3}}, & x \neq -1, \\ a, & x = -1,\end{cases} \quad x_{0} = -1 ;$

(3)

$f(x) = \begin{cases} a x^{2}+1, & x \neq 0, \quad x_{0} = 0 ; \\ -x, & x \leqslant 0,\end{cases}$

(4)

$f(x) = \begin{cases} \cos x, & x \leqslant 0, \\ a(x-1), & x > 0, \quad x_{0} = 0.\end{cases}$

Problem 1.10.20

(1)(2)(3)(4)

Determine whether values of $a$ and $b$ exist for which the function $f$ is continuous on its domain, if:

?

(1)

$f(x) = \begin{cases} (x-1)^{3}, & x \leqslant 0 \\ a x+b, & 0 < x < 1 \\ \sqrt{x}, & x \geqslant 1\end{cases}$

(2)

$f(x) = \begin{cases} x, & \left|x\right| \leqslant 0, \\ x^{2}+a x+b, & \left|x\right| > 1\end{cases}$

(3)

$f(x) = \begin{cases} \frac{(x-1)^{2}}{x^{2}-1}, & \left|x\right| \neq 1 \\ a, & x = -1 \\ b, & x = 1\end{cases}$

(4)

$f(x) = \begin{cases} \frac{x \cos (x / 2)}{\sin x}, & x \in \left[-\frac{\pi }{2} ; \frac{3 \pi }{2}\right], x \neq 0, \quad x \neq \pi , \\ a, & x = 0, \\ b, & x = \pi .\end{cases}$

Problem 1.10.21

Prove that if a function is monotone, then every discontinuity of it is a discontinuity of the first kind.

?

Problem 1.10.22

Prove that the function

f(x) = \begin{cases} 1, & \text{ if } & x \text{ is a rational number, } \\ 0, & \text{ if } & x \text{ is an irrational number, } \end{cases}

is discontinuous at every point.

?

Problem 1.10.23

Prove that the function

f(x) = \begin{cases} x, & \text{ if } & x \text{ is a rational number, } \\ 0, & \text{ if } & x \text{ is an irrational number } \end{cases}

is continuous at the point $x = 0$ and discontinuous at every other point.

?

Problem 1.10.24

Prove the continuity of the function at every point of its domain:

?

(1)

$y = 3 x^{5}+1 / x^{3}$

(2)

$y = \operatorname {sh} x$

(3)

$y = \sqrt[3]{x^{2}-6 x^{3}}$

(4)

$y = \frac{\sin x^{2}}{x^{4}+2 x^{2}}$ ;

(5)

$y = \cos \left(x-\sqrt{1-x^{2}}\right)$ ;

(6)

$y = x e^{(\sin x) / x}$ .

Problem 1.10.25

(1)(2)(3)(4)

Investigate the continuity of the functions $f(g(x))$ and $g(f(x))$ at the points where these compositions are defined, if:

?

(1)

$f(x) = \operatorname {sign} x, g(x) = 1+x^{2}$ ;

(2)

$f(x) = \operatorname {sign} x, g(x) = x^{3}-x$ ;

(3)

$f(x) = \operatorname {sign}(x-1), g(x) = \operatorname {sign}(x+1)$ ;

(4)

$f(x) = \operatorname {sign} x, g(x) = 1+x-\mathbb {E}\left[x\right]$ .

Problem 1.10.26

Prove that a polynomial is a function continuous at every point.

?

Problem 1.10.27

Prove that the function $y = \frac{P(x)}{Q(x)}$ , where $P$ and $Q$ are nonzero polynomials, is continuous at every point $x_{0}$ where $Q\left(x_{0}\right) \neq 0$ .

?

Problem 1.10.28

(1)(2)

?

(1)

The function $f$ is continuous at the point $x_{0}$ , while the function $g$ is discontinuous at the point $x_{0}$ . Prove that the function $f+g$ is discontinuous at this point.

(2)

Give an example of functions $f$ and $g$ , discontinuous at the point $x_{0}$ , whose sum is: а) discontinuous at the point $x_{0}$ ; б) continuous at the point $x_{0}$ .

Problem 1.10.29

(1)(2)

?

(1)

Give an example of a function $f$ continuous at the point $x_{0}$ and a function $g$ discontinuous at the point $x_{0}$ , whose product is: a) discontinuous at the point $x_{0}$ ; б) continuous at the point $x_{0}$ .

(2)

Give an example of functions $f$ and $g$ , discontinuous at the point $x_{0}$ , whose product is: а) discontinuous at the point $x_{0}$ , б) continuous at the point $x_{0}$ .

Problem 1.10.30

Give an example of functions $f$ and $g$ , continuous at the point $x_{0}$ , whose quotient $f / g$ is discontinuous at the point $x_{0}$ .

?

Problem 1.10.31

Prove the substitution rule for limits of continuous functions: let the function $y = g(x)$ be continuous at the point $x_{0}$ , and let the function $f(y)$ be continuous at the point $y_{0} = g\left(x_{0}\right)$ ; then

\lim _{x \rightarrow x_{0}} f(g(x)) = \lim _{y \rightarrow y_{0}} f(y)

?

Problem 1.10.32

Prove that the limit sign and the sign of a continuous function commute: let the function $y = g(x)$ have a limit $\lim_{x \rightarrow x_{0}} g(x) = y_{0}$ , and let the function $f(y)$ be continuous at the point $y_{0}$ ; then, in some neighborhood of the point $x_{0}$ , excluding perhaps this point itself, the composition $f(g(x))$ is defined and there exists

\lim _{x \rightarrow x_{0}} f(g(x)) = f\left(\lim _{x \rightarrow x_{0}} g(x)\right) = f\left(y_{0}\right)

?

Problem 1.10.33

(1)(2)(3)

Starting from the continuity of the exponential function, prove the continuity of the hyperbolic functions:

?

(1)

$y = \operatorname {ch} x$

(2)

$y = \operatorname {th} x$

(3)

$y = \operatorname {cth} x$ .

Problem 1.10.34

Starting from the continuity of the exponential function, prove the continuity of the logarithmic function $y = \log_{a} x, a > 0, a \neq 1$ .

?

Problem 1.10.35

(1)(2)(3)(4)

Starting from the continuity of the trigonometric functions, prove the continuity of the inverse trigonometric functions:

?

(1)

$y = \arcsin x$

(2)

$y = \arccos x$ ;

(3)

$y = \operatorname {arctg} x$ ;

(4)

$y = \operatorname {arcctg} x$ .

Problem 1.10.36

Starting from the continuity of the exponential function, prove the continuity of the power function $y = x^{\mu }(x > 0, \mu \in \mathbb {R})$ .

?

Problem 1.10.37

Prove that if $y = f(x)$ is a continuous function, then the functions $y = \left|f(x)\right|, y = f(\left|x\right|)$ are also continuous.

?

Problem 1.10.38

Let $f$ be a function continuous on an interval $X$ . Prove that the functions

$f_{+}(x) = \begin{cases} f(x), & \text{ if } f(x) > 0, \\ 0, & \text{ if } f(x) \leqslant 0,\end{cases} \quad f_{-}(x) = \begin{cases} 0, & \text{ if } f(x) \geqslant 0, \\ f(x), & \text{ if } f(x) < 0,\end{cases}$ are continuous on the interval $X$ .

?

Problem 1.10.39

Give an example of a function continuous on each of the intervals $X_{1}$ and $X_{2}$ , but not continuous on the set $X_{1} \cup X_{2}$ .

?

Problem 1.10.40

(1)(2)

Give an example of a function continuous on an interval:

?

(1)

unbounded on this interval;

(2)

bounded on this interval, but attaining neither its supremum nor its infimum.

Problem 1.10.41

(1)(2)(3)

?

(1)

A function $f$ is defined and continuous on the closed interval $[a ; b]$ , and all its values are positive. Prove that there exists a number $\mu > 0$ such that $f(x) \geqslant \mu$ for any $x \in [a ; b]$ .

(2)

Give an example of a function $f$ , continuous on the open interval $(a ; b)$ , taking only positive values and such that for any $\mu > 0$ there is a value $f(x) < \mu , x \in (a ; b)$ .

(3)

Give an example of a function $f$ , defined on the closed interval $[a ; b]$ , taking only positive values and such that for any $\mu > 0$ there is a value of the function $f(x) < \mu , x \in (a ; b)$ .

Problem 1.10.42

The function $f$ is continuous on the interval $(a ; b)$ ,

m = \inf _{(a, b)} f, \quad M = \sup _{(a, b)} f

Prove that for any $y \in (m ; M)$ there exists $x \in (a ; b)$ such that $f(x) = y$ .

?

Problem 1.10.43

Prove that if a function is defined and continuous on a closed interval, then its range is a closed interval.

?

Problem 1.10.44

Give an example of a discontinuous function, defined on a closed interval, whose range is a closed interval.

?

Problem 1.10.45

Give an example of a continuous function that takes the values 1 and 3, but does not take the value 2.

?

Problem 1.10.46

Let a function be defined and monotone on an interval, and let its range be an interval. Prove that this function is continuous.

?

Problem 1.10.47

(1)(2)

Prove that the equation $x^{5}-3 x = 1$ :

?

(1)

has at least one root on $(1 ; 2)$ ;

(2)

has at least three roots on $\mathbb {R}$ .

Problem 1.10.48

(1)(2)

Prove that:

?

(1)

any polynomial of odd degree has at least one real root;

(2)

if a polynomial of even degree takes at least one value opposite in sign to the coefficient of its leading term, then it has at least two real roots.

Problem 1.10.49

Prove that the equation $x = y-\varepsilon \sin y$ , where $0 < \varepsilon \leqslant 1$ , defines a single continuous function $y = f(x)$ .

?

Problem 1.10.50

Prove that the discontinuous function $y = e^{\left|x\right|} \operatorname {sign} x$ has a continuous inverse.

?

Problem 1.10.51

(1)(2)(3)

Prove that the given system of equations defines a continuous function $y(x)$ or $x(y)$ ; plot the graph of this function:

?

(1)

$x = \operatorname {arctg} t, y = \frac{1}{1+t^{2}}$ ;

(2)

$x = \operatorname {ch} t, y = \operatorname {sh} t$ ;

(3)

$x = \ln \left(1+e^{-t}\right), y = \ln \left(1+e^{t}\right)$ .

Problem 1.10.52

Prove that the system of equations

x = \left(t^{2}+1\right) / 2 t, \quad y = \left(t^{2}-1\right) / 2 t

defines two functions $x(y)$ , continuous on $\mathbb {R}$ , and four functions $y(x)$ , continuous on the set $\left\{ x:\left|x\right| \geqslant 1\right\}$ .

?

Problem 1.10.53

Let the function $f$ be defined in a neighborhood of the point $x_{0}$ . Will the function $f$ be continuous at the point $x_{0}$ , if:

?

(1)

$\exists \varepsilon > 0 \quad \exists \delta > 0 \quad \forall x \quad \left(\left|x-x_{0}\right| < \delta \Rightarrow \left|f(x)-f\left(x_{0}\right)\right| < \varepsilon \right)$ ;

(2)

$\forall \delta > 0 \quad \exists \varepsilon > 0 \quad \forall x \quad \left(\left|x-x_{0}\right| < \delta \Rightarrow \left|f(x)-f\left(x_{0}\right)\right| < \varepsilon \right)$ ;

(3)

$\forall \varepsilon > 0 \exists \delta > 0 \forall x \quad \left(\left|f(x)-f\left(x_{0}\right)\right| < \varepsilon \Rightarrow \left|x-x_{0}\right| < \delta \right)$ ;

(4)

$\forall \delta > 0 \quad \exists \varepsilon > 0 \quad \forall x \quad \left(\left|f(x)-f\left(x_{0}\right)\right| < \varepsilon \Rightarrow \left|x-x_{0}\right| < \delta \right)$ ;

(5)

$\exists \varepsilon > 0 \forall \delta > 0 \forall x \quad \left(\left|x-x_{0}\right| < \delta \Rightarrow \left|f(x)-f\left(x_{0}\right)\right| < \varepsilon \right)$ ?

Problem 1.10.54

The function $f$ is defined in a neighborhood of the point $x_{0}$ , and there exists a sequence $\left\{ \varepsilon_{n}\right\}$ such that $\varepsilon_{n} > 0, n \in \mathbb {N}, \lim_{n \rightarrow \infty } \varepsilon_{n} = 0$ , and for each $\varepsilon_{n}$ there exists $\delta_{n} > 0$ such that if $x \in D(f)$ and $\left|x-x_{0}\right| < \delta_{n}$ , then $\left|f(x)-f\left(x_{0}\right)\right| < \varepsilon_{n}$ . Prove that the function $f$ is continuous at the point $x_{0}$ .

?

Problem 1.10.55

The function $f$ is continuous at the point $x_{0}$ . Let

S(\delta ) = \sup _{\left(x_{0}-\delta , x_{0}+\delta \right)} f, \quad s(\delta ) = \inf _{\left(x_{0}-\delta , x_{0}+\delta \right)} f

Prove that $\lim_{\delta \rightarrow 0}(S(\delta )-s(\delta )) = 0$ .

?

Problem 1.10.56

Indicate the set of points at which the function is continuous, find its points of discontinuity, determine their type, and draw the graph of the function:

?

(1)

$y = \begin{cases} x^{2}+2, & x \leqslant 0 \\ x-1, & x > 0 \end{cases}$

(2)

$y = \begin{cases} 1-x^{3}, & x < 0 \\ (x-1)^{3}, & 0 \leqslant x \leqslant 2, \\ 4-x, & 2 < x\end{cases}$

(3)

$y = \begin{cases} -1 / x, & x < 0 \\ 5 x-x^{2}, & x \geqslant 0\end{cases}$

(4)

$y = \begin{cases} -x, & x \leqslant -1 \\ 2 /(x-1), & x > -1\end{cases}$

(5)

$y = \begin{cases} 2^{x}, & -1 \leqslant x < 1 \\ 1, & x = 1 \\ x-1, & 1 < x \leqslant \infty \end{cases}$

(6)

$y = \begin{cases} \cos x, & -\pi / 2 \leqslant x < \pi / 4 \\ 1, & x = \pi / 4 \\ x^{2}-\pi^{2} / 16, & \pi / 4 < x \leqslant \pi \end{cases}$

Problem 1.10.57

Find the points of discontinuity of the function, determine their type, and extend the function by continuity at the points of removable discontinuity.

?

(1)

$y = \frac{x}{x^{2}+x-6}$ ;

(2)

$y = \frac{1}{x^{3}-3 x^{2}-4 x}$ ;

(3)

$y = \frac{1}{\sqrt{x^{2}-1}}$ ;

(4)

$y = \frac{1+x}{1+x^{3}}$

(5)

$y = \frac{1 / x-1 /(x+1)}{1 /(x-1)-1 / x}$

(6)

$y = \frac{2 x-1}{2 x^{2}+3 x-2}$

(7)

$y = \frac{1-\sqrt{x}}{x^{2}-1}$ .

Problem 1.10.58

Find the points of discontinuity of the function, determine their type, and extend the function by continuity at the points of removable discontinuity.

?

(1)

$y = \frac{x}{\cos x}$ ;

(2)

$y = (\sin x) \sin \frac{1}{x}$

(3)

$y = \frac{\arcsin x}{\sin 2 x}$

(4)

$y = \frac{\cos (\pi x / 2)}{x^{3}-x^{2}}$ ;

(5)

$y = \frac{\sin 3 x}{\sin 2 x}$ .

Problem 1.10.59

Find the points of discontinuity of the function, determine their type, and extend the function by continuity at the points of removable discontinuity.

?

(1)

$y = \frac{2}{1-2^{x}}$ ;

(2)

$y = \frac{1}{\lg x}$ ;

(3)

$y = \lg \left(x^{2}+3 x\right)$ ;

(4)

$y = \lg (x-1)^{2}$ ;

(5)

$y = 2^{1 / x}$ ;

(6)

$y = \frac{1}{\ln \left|x-1\right|}$

(7)

$y = 3^{x /\left(1-x^{2}\right)}$ ;

(8)

$y = e^{-1 /\left|x\right|}$

(9)

$y = \ln \ln \left(1+x^{2}\right)$ .

Problem 1.10.60

Find the points of discontinuity of the function, determine their type, and find the jumps at the discontinuities of the first kind:

?

(1)

$y = \operatorname {sign}\left(x^{2}-2 x-3\right)$

(2)

$y = \operatorname {sign} \cos x$

(3)

$y = (-1)^{\mathbb {E}\left[x\right]}$

(4)

$y = (-1)^{E(1 / x)}$ ;

(5)

$y = \arcsin (1 / x)$ ;

(6)

$y = \operatorname {arctg}(1 / x)$ ;

(7)

$y = \operatorname {arcctg}\left(\frac{1}{x^{2}}\right)$ ;

(8)

$y = \frac{x+1}{\operatorname {arctg}(1 / x)}$

(9)

$y = \frac{\left|x\right|}{\operatorname {arctg} x}$

(10)

$y = \frac{1}{1+2^{1 /(x-1)}}$ ;

(11)

$y = \frac{3^{1 / x}+2^{1 / x}}{3^{1 / x}-2^{1 / x}}$

(12)

$y = \frac{1}{x} \ln \frac{1+x}{1-x}$ .

Problem 1.10.61

(1)(2)(3)

Prove that the function is continuous at every point of its domain:

?

(1)

$y = \sin (x-\lg (\sqrt{x}-1))$

(2)

$y = \sqrt[3]{x^{2}-4 \operatorname {ctg} x} 2^{\operatorname {tg}(\pi / x)}$ ;

(3)

$y = \frac{x^{-5 / 3}-\cos \sqrt{x}}{\operatorname {tg} \arcsin \left|x\right|}$ .

Problem 1.10.62

Find the value of $a$ for which the function $y(x)$ will be continuous, if:

?

(1)

$y = \begin{cases} \frac{(1+x)^{n}-1}{x}, & x \neq 0, \quad n \in \mathbb {N}, \\ a, & x = 0 ;\end{cases}$

(2)

$y = \begin{cases} x \operatorname {ctg} 2 x, & x \neq 0, \quad \left|x\right| < \pi / 2, \\ a, & x = 0 ;\end{cases}$

(3)

$y = \begin{cases} (\pi +2 x) \operatorname {tg} x, & -\pi < x < \pi / 2, \quad x \neq -\pi / 2, \\ a, & x = -\pi / 2\end{cases}$

(4)

$y = \begin{cases} (\arcsin x) \operatorname {ctg} x, & x \neq 0, \\ a, & x = 0,\end{cases}$

(5)

$y = \begin{cases} \frac{c^{x}-1}{x}, & x \neq 0, \\ a, & x = 0, \quad c > 0\end{cases}$

(6)

$y = \begin{cases} \frac{x}{\ln (1+2 x)}, & x \neq 0, \\ a, & x = 0 ;\end{cases}$

(7)

$y = \begin{cases} \frac{\operatorname {sh} x}{x}, & x \neq 0, \\ a, & x = 0\end{cases}$

(8)

$y = \begin{cases} x \ln x^{2}, & x \neq 0 \\ a, & x = 0\end{cases}$

(9)

$y = \begin{cases} e^{-1 / x^{2},} & x \neq 0 \\ a, & x = 0\end{cases}$

(10)

$y = \begin{cases} (1+x)^{1 / x}, & x \neq 0, \\ a, & x = 0.\end{cases}$

Problem 1.10.63

Determine whether the function can be extended at the point of discontinuity $x_{0}$ so that it becomes continuous at this point, if:

?

(1)

$y = \frac{1}{x}+\frac{1}{\left|x\right|}, x_{0} = 0$

(2)

$y = 2^{-2^{1 /(1-x)}}, x_{0} = 1$ ;

(3)

$y = \frac{1}{x} e^{-1 / x^{2}}, x_{0} = 0$ ;

(4)

$y = \frac{\operatorname {th}(\sqrt{x}-1)}{x^{2}-1}, x_{0} = 1$ ;

(5)

$y = 2^{-E(1 / x)}, x_{0} = 0$ .

Problem 1.10.64

Investigate the continuity and plot the graph of the function $f$ , if:

?

(1)

$f(x) = \lim_{n \rightarrow \infty } \frac{x^{2 n}-1}{x^{2 n}+1}$ ;

(2)

$f(x) = \lim_{n \rightarrow \infty } \frac{1}{1+x^{2 n+1}}$

(3)

$f(x) = \lim_{n \rightarrow \infty } \cos^{n} x$

(4)

$f(x) = \lim_{n \rightarrow \infty } \frac{x}{1+(2 \sin x)^{2 n}}$

(5)

$f(x) = \lim_{n \rightarrow \infty } \frac{n^{x}-n^{-x}}{n^{x}+n^{-x}}$

(6)

$f(x) = \lim_{n \rightarrow \infty } \frac{x+e^{n x}}{1+x e^{n x}}$

(7)

$f(x) = \lim_{n \rightarrow \infty } \sqrt[n]{1+x^{2 n}}$ ;

(8)

$f(x) = \lim_{n \rightarrow \infty }(x \operatorname {arctg}(n \operatorname {ctg} x))$ ;

(9)

$f(x) = \lim_{t \rightarrow +\infty } \frac{\ln \left(1+e^{x t}\right)}{\ln \left(1+e^{t}\right)}$ ;

(10)

$f(x) = \lim_{t \rightarrow +\infty }(1+x) \operatorname {tg} x t$ .

Problem 1.10.65

At which points is the function continuous

y(x) = \begin{cases} x^{2}-1, & x \text{ is irrational, } \\ 0, & x \text{ is rational? }\end{cases}

?

Problem 1.10.66

(1)(2)

Let

y(x) = \begin{cases} 0, & x \text{ is irrational, } \\ 1 / q, & x = p / q, \quad p \in \mathbb {Z}, \quad q \in \mathbb {N}\end{cases}

where $p / q$ is in lowest terms (this function is called the Riemann function).

Prove that:

?

(1)

this function is continuous at every irrational point;

(2)

every rational point is a discontinuity of the first kind for this function.

Problem 1.10.67

Investigate the continuity of the function

y(x) = \begin{cases} \left|x\right|, & x \text{ is irrational, } \\ q x /(q+1), & x = p / q, \quad p \in \mathbb {Z}, \quad q \in \mathbb {N}\end{cases}

where $p / q$ is in lowest terms.

?

Problem 1.10.68

Investigate the continuity of the compositions $f \circ g$ and $g \circ f$ , if

y(x) = \begin{cases} x, & x \text{ is a rational number, } \\ 2-x, & x \text{ is an irrational number. }\end{cases}

?

Problem 1.10.69

(1)(2)

The function $f$ is increasing on the closed interval $[a ; b]$ and discontinuous at the point $x_{0} \in [a ; b]$ . The function $g(x)$ is monotone on the closed interval $[f(a) ; f(b)]$ .

?

(1)

Give an example of such functions $f$ and $g$ that $g(f(x))$ is continuous at $x_{0}$ .

(2)

Prove that if $g(x)$ is strictly monotone in a neighborhood of the point $f\left(x_{0}\right)$ , then $g(f(x))$ is discontinuous at the point $x_{0}$ .

Problem 1.10.70

(1)(2)

The function $f$ is continuous on the closed interval $[a ; b]$ . Prove that:

?

(1)

$\inf_{(a ; b)} f = \inf_{[a ; b]} f$ ;

(2)

$\sup_{(a ; b)} f = \sup_{[a, b]} f$ .

Problem 1.10.71

(1)(2)

The function $f$ is continuous on the closed interval $[a ; b]$ . Prove that the functions $m(x)$ and $M(x)$ are continuous on $[a ; b]$ , if:

?

(1)

$m(x) = \min_{[a ; x]} f$ ;

(2)

$M(x) = \max_{[a ; x]} f$ .

Problem 1.10.72

(1)(2)

The function $f$ is continuous on the interval $[a ; b)$ . Prove that the functions $m(x)$ and $M(x)$ are continuous on $(a, b)$ , if:

?

(1)

$m(x) = \inf_{[a ; x)} f$ ;

(2)

$M(x) = \sup_{[a ; x)} f$ .

Problem 1.10.73

The function $f$ is defined and bounded on the closed interval $[a ; b]$ . Prove that the functions

m(x) = \inf _{[a ; x]} f \quad \text{ and } \quad M(x) = \sup _{[a, x)} f

are continuous from the left at every point $x \in (a ; b]$ .

?

Problem 1.10.74

(1)(2)

Let $f$ and $g$ be functions continuous on $X$ . Prove that the functions $M(x)$ and $m(x)$ are also continuous on $X$ , if:

?

(1)

$M(x) = \max \left\{ f(x), g(x)\right\}$ ;

(2)

$m(x) = \min \left\{ f(x), g(x)\right\}$ .

Problem 1.10.75

(1)(2)

?

(1)

Let $f$ be a function continuous on $X$ , $a, b \in \mathbb {R}, a < b$ . Prove that the function

f(a ; b ; x) = \begin{cases} f(x), & \text{ if } & a \leqslant f(x) \leqslant b, \\ a, & \text{ if } & f(x) < a \\ b, & \text{ if } & f(x) > b \end{cases}

is also continuous on $X$ .

(2)

Let the function $f$ be defined on $\mathbb {R}$ . Prove that for $f$ to be continuous on $\mathbb {R}$ , it is necessary and sufficient that for any $a > 0$ the function $f(-a ; a ; x)$ , defined in 1), be continuous on $\mathbb {R}$ .

Problem 1.10.76

The function $f$ is continuous on $[a ;+\infty )$ , and $\lim_{x \rightarrow +\infty } f(x)$ exists and is finite. Prove that $f$ is bounded on $[a ;+\infty )$ .

?

Problem 1.10.77

The function $f$ is continuous on the interval $(a ; b)$ (finite or infinite), and $\lim_{x \rightarrow a+0} f(x)$ and $\lim_{x \rightarrow b-0} f(x)$ exist and are finite. Prove that the function $f$ is bounded on $(a ; b)$ .

?

Problem 1.10.78

(1)(2)

?

(1)

Prove that the function

y = \begin{cases} (x+1) 2^{-(1 /\left|x\right|+1 / x)}, & x \neq 0, \quad \left|x\right| \leqslant 2 \\ 0, & x = 0\end{cases}

takes all values between $f(-2)$ and $f(2)$ , but is discontinuous. Plot the graph of this function.

(2)

Prove that the function

f(x) = \begin{cases} \sin (1 / x), & x \neq 0 \\ -1, & x = 0\end{cases}

takes, on every closed interval $[0 ; a]$ , all intermediate values between $f(0)$ and $f(a)$ , but is not continuous on $[0 ; a]$ .

Problem 1.10.79

(1)(2)

The function $f$ is defined on the closed interval $[a ; b]$ and has the following property: for any $x_{1}, x_{2} \in [a ; b]$ and any number $C$ lying between $f\left(x_{1}\right)$ and $f\left(x_{2}\right)$ , there exists a point $\xi \in \left(x_{1} ; x_{2}\right)$ such that $f(\xi ) = C$ .

?

(1)

Indicate a function having this property but not continuous on $[a ; b]$ .

(2)

Prove that a function having this property cannot have discontinuities of the first kind.

Problem 1.10.80

Prove that if a function is defined and continuous on an interval, then its range is an interval (i.e., a closed interval, an open interval, or a half-open interval).

?

Problem 1.10.81

(1)(2)(3)

Give an example of a function, continuous on an interval, whose range is:

?

(1)

a closed interval;

(2)

an open interval;

(3)

a half-open interval.

Problem 1.10.82

Let a function, defined on the closed interval $[a ; b]$ , be continuous and invertible. Prove that this function is strictly monotone on $[a ; b]$ .

?

Problem 1.10.83

Prove that if a function is defined and strictly monotone on an interval, then its inverse function is continuous.

?

Problem 1.10.84

Give an example of a function $f$ , continuous at the point $x_{0}$ , whose inverse function is discontinuous at the point $y_{0} = f\left(x_{0}\right)$ .

?

Problem 1.10.85

Give an example of a continuous, strictly increasing function whose inverse is discontinuous.

?

Problem 1.10.86

Prove that a function bounded and convex (see problem 237, §7) on an interval is continuous on this interval.

?

Problem 1.10.87

(1)(2)(3)

Prove that the given system of equations defines a continuous function $y(x)$ or $x(y)$ :

?

(1)

$x = \frac{t-1}{\sqrt{t}}, y = \frac{t+1}{\sqrt{t}}$

(2)

$x = \frac{1}{4}(t-4) e^{t}, y = \sqrt{t} e^{t}$

(3)

$x = \frac{t^{3}}{t^{2}+1}, y = \frac{t}{t^{4}+1}$ .

Problem 1.10.88

Give an example of a system of equations

x = \varphi (t), \quad y = \psi (t)

in which the functions $\varphi$ and $\psi$ are not invertible, but which defines a unique continuous function $y(x)$ .

?

Problem 1.10.89

The function $f$ is continuous on the interval $(a ; b)$ . Prove that for any numbers $x_{1} < x_{2} < \ldots < x_{n}$ from $(a ; b)$ and any numbers

\alpha _{i} \geqslant 0, \quad \sum _{i = 1}^{n} \alpha _{i} = 1

there is a number $\xi , x_{1} \leqslant \xi \leqslant x_{n}$ , such that $f(\xi ) = \sum_{i = 1}^{n} \alpha_{i} f\left(x_{i}\right)$ .

?

Problem 1.10.90

A function $f$ is defined on the closed interval $[a ; b]$ . For every closed interval $[c ; d], a < c < d < b$ , the set of values $f(x), x \in [c ; d]$ , is a closed interval. Does it follow from this that the function $f$ is continuous on $[a ; b]$ ?

?

Problem 1.10.91

The function $f$ is continuous and bounded on the interval $(a ;+\infty )$ and has no limit as $x$ tends to $+\infty$ . Prove that there is a number $a$ for which the equation $f(x) = a$ has infinitely many solutions.

?

Problem 1.10.92

The function $f$ is continuous on the interval $(a ; b), l = \varliminf_{x \rightarrow b} f(x), L = \varlimsup_{x \rightarrow b} f(x), L > l$ . Prove that for any number $C, l < C < L$ , the equation

f(x) = C

has infinitely many solutions in any neighborhood of $b$ .

?

Problem 1.10.93

The function $f$ is continuous and bounded on the interval $[a ;+\infty )$ . Prove that for any number $T$ there is a sequence $\left\{ x_{n}\right\}$ such that $\lim_{n \rightarrow \infty } x_{n} = +\infty$ and

\lim _{n \rightarrow \infty }\left(f\left(x_{n}+T\right)-f\left(x_{n}\right)\right) = 0

?

Problem 1.10.94

(1)(2)

A set $A \in \mathbb {R}$ is called open if every point of $A$ has a neighborhood belonging to $A$ . A point $x_{0}$ is called a point of closure of the set $A$ if every neighborhood of $x_{0}$ contains at least one point of $A$ . A set is called closed if it contains all its points of closure. The set of all points of closure of the set $A$ is called the closure of $A$ and is denoted by $\bar{A}$ .

The function $f$ is continuous on the closed interval $[a ; b]$ , and the number $C$ lies between $f(a)$ and $f(b), f(a) \neq f(b)$ .

Prove that:

?

(1)

each of the sets is open: а) $A = \left\{ x \in (a ; b): f(x) < C\right\}$ , б) $B = \left\{ x \in (a ; b): f(x) > C\right\}$ ;

(2)

the set $\left\{ x \in [a ; b]: f(x) = C\right\}$ has both a largest and a smallest element.

Problem 1.10.95

(1)(2)

Let the function $f$ be continuous on $R$ , $A \subset E(f)$ (the set of values of $f$ ), $f^{-1}(A) = \left\{ x \in \mathbb {R} \mid f(x) \in A\right\}$ . Prove that:

?

(1)

if $A$ is closed, then $f^{-1}(A)$ is also closed;

(2)

if $A$ is open, then $f^{-1}(A)$ is also open.

Problem 1.10.96

A set $B$ is called dense in a set $A$ if the closure of $B$ contains $A$ , i.e., $\bar{B} \supset A$ . Let the function $f$ be continuous on $X$ , let the set $A$ be dense in $X$ , let $f(X)$ be the set of all values of the function $f(x)$ for $x \in X$ , and let $f(A)$ be the set of all values of the function for $x \in A$ . Prove that $f(A)$ is dense in $f(X)$ .

?

Problem 1.10.97

(1)(2)

?

(1)

Does there exist a continuous map: а) of a closed interval onto an open interval; б) of an open interval onto a closed interval?

(2)

Construct a one-to-one map of a closed interval onto an open interval.

Problem 1.10.98

(1)(2)

?

(1)

Prove that a function defined on $\mathbb {R}$ cannot be continuous at all rational points and discontinuous at all irrational points.

(2)

Does there exist a function continuous at all rational points of the closed interval $[0 ; 1]$ and discontinuous at all its irrational points?

Problem 1.10.99

Prove that the equation $x^{3}-3 x^{2}+6 x-1$ has a unique root; find this root to within $0,1$ .

?

Problem 1.10.100

Prove that the equation $x^{4}+3 x^{2}-x-2 = 0$ has two (and no more) real roots.

?

Problem 1.10.101

Prove that the equation

\sum _{k = 0}^{n} a_{k} x^{2 k+1}+c = 0

where $a_{k} \geqslant 0(k = 0,1, \ldots , n-1), a_{n} > 0$ , has only one real root.

?

Problem 1.10.102

Prove that the equation $x^{n} = P_{n-1}(x)$ , where $P_{n-1}(x)$ is a polynomial of degree $(n-1)$ with positive coefficients, has a unique positive root.

?

Problem 1.10.103

(1)(2)

Prove that:

?

(1)

the equation

\frac{a_{1}}{x-\lambda _{1}}+\frac{a_{2}}{x-\lambda _{2}}+\frac{a_{3}}{x-\lambda _{3}} = 0

where $a_{1} > 0, a_{2} > 0, a_{3} > 0, \lambda_{1} < \lambda_{2} < \lambda_{3}$ , has exactly one real root in each of the intervals $\left(\lambda_{1} ; \lambda_{2}\right)$ and $\left(\lambda_{2} ; \lambda_{3}\right)$ ;

(2)

the equation

\sum _{j = 1}^{n} \frac{a_{\jmath }}{x-\lambda _{\jmath }} = 0

where $a_{j} > 0(j = 1,2, \ldots , n), \quad \lambda_{1} < \lambda_{2} < \ldots < \lambda_{n}$ , has exactly one real root in each of the intervals $\left(\lambda_{j} ; \lambda_{\jmath+1}\right)(j = 1,2, \ldots , n-1)$ .

Problem 1.10.104

(1)(2)(3)(4)

Prove that the equation has a solution, and that it is unique:

?

(1)

$x \cdot 2^{x} = 1$ ;

(2)

$x e^{x} = 2$ ;

(3)

$x = \varepsilon \sin x+a, 0 < \varepsilon < 1$ ;

(4)

$x^{2} \operatorname {arctg} x = a, a \neq 0$ .

Problem 1.10.105

Prove that the equation $x = a \sin x+b$ , where $0 < a < 1, b > 0$ , has at least one root on $[0 ; a+b]$ .

?

Problem 1.10.106

Prove that the equation $10^{x-1} = x$ has only one root $x_{0} \neq 1$ .

?

Problem 1.10.107

Prove that the equation $2^{x} = 4 x$ has at least two real roots.

?

Problem 1.10.108

Prove that the equation $x \sin x-0,5 = 0$ has infinitely many solutions.

?

Problem 1.10.109

(1)(2)(3)(4)

?

(1)

Prove that there exist infinitely many functions, defined on $(a ; b)$ , satisfying the equation

y^{2} = 1

(2)

Let $f$ be a function continuous and positive on $(a ; b)$ . Prove that there exists a unique continuous function $y = \varphi (x)$ on $(a ; b)$ satisfying the equation

y^{2} = f(x)

and the condition that $\varphi \left(x_{0}\right) > 0$ at some point $x_{0} \in (a ; b)$ .

(3)

How many functions, continuous on $\mathbb {R}$ , satisfy the equation

y^{2} = \left(x^{2}-1\right)^{2}?

(4)

How many functions, continuous on the closed interval $[0 ; n], n \in \mathbb {N}$ , satisfy the equation

y^{2}+2 y+\cos ^{2} \pi x = 0?

Problem 1.10.110

Prove that there exists a unique continuous function $y(x)$ satisfying the equation

y = \operatorname {arcctg} x y, \quad x \in \mathbb {R}

?

Problem 1.10.111

Find all functions $f$ , continuous on $\mathbb {R}$ , satisfying the condition: for any $x \in \mathbb {R}$ the equality holds:

?

(1)

$f(x / 3)+f(2 x / 3) = x$ ;

(2)

$f(x)+f(3 x) = x$ ;

(3)

$f(\alpha x)+f(\beta x) = a x+b, \alpha \beta \neq 0$ ;

(4)

$f(3 x)-f(2 x) = x$ ;

(5)

$f\left(x^{2}\right)+f(x) = x^{2}+x$ ;

(6)

$f\left(x^{2}\right)-f(x) = x^{2}-x$ .

Problem 1.10.112

Find all functions $f$ , continuous on $\mathbb {R}$ , such that for any $x \in \mathbb {R}$ the equality holds

A f(\alpha x)+B f(\beta x) = a x+b,

where $a \alpha \beta A B \neq 0, A \alpha +B \beta \neq 0, A+B \neq 0$ .

?

Problem 1.10.113

(1)(2)

For which а do there exist functions $f$ , continuous on $\mathbb {R}$ and not identically constant, such that for any $x \in \mathbb {R}$ :

?

(1)

$f\left(x^{2}+a\right) = f(x)$ ;

(2)

$f\left(a x^{2}+1\right) = f(x)$ ?

Problem 1.10.114

For which $a, b, c$ does there exist a function $f$ , continuous on $\mathbb {R}$ and not identically constant, such that for any $x \in \mathbb {R}$

f\left(a x^{2}+b x+c\right) = f(x)?

?

Problem 1.10.115

(1)(2)

?

(1)

Find all functions $f$ , continuous on $\mathbb {R}$ , satisfying the equality $f(x+y) = f(x)+f(y)$ for any $x, y \in \mathbb {R}$ .

(2)

Give an example of a discontinuous function $f$ satisfying the equality $f(x+y) = f(x)+f(y)$ for any $x, y \in \mathbb {R}$ .

Problem 1.10.116

Find all functions, continuous on $\mathbb {R}$ , satisfying the equality $f(x+y) = f(x) f(y)$ for any $x, y \in \mathbb {R}$ .

?

Problem 1.10.117

(1)(2)

Find all functions, continuous on $(0 ;+\infty )$ , satisfying the equality for any $x, y \in (0 ;+\infty )$ :

?

(1)

$f(x y) = f(x)+f(y)$ ;

(2)

$f(x y) = f(x) f(y)$ .

Problem 1.10.118

(1)(2)

Find all functions, continuous on $\mathbb {R}$ , satisfying the equality for any $x, y \in \mathbb {R}$

f(x+y)+f(x-y) = 2 f(x) f(y)

and the condition:

?

(1)

$f(x) \leqslant 1$ ;

(2)

$f(x) \geqslant 1$ .

Problem 1.10.119

Find all solutions, continuous on $\mathbb {R}$ , of the functional equation

f(x+f(x)) = f(x), \quad x \in \mathbb {R}

?

Problem 1.10.120

The function $f$ is defined and continuous on $\mathbb {R}$ , and for any $x, y \in \mathbb {R}$ the equality holds

f\left(\frac{x+y}{2}\right) = \frac{1}{2}(f(x)+f(y))

Prove that $f$ is a linear function, i.e., there exist $a$ and $b$ such that $f(x) = a x+b$ .

?

Problem 1.10.121

Prove that a periodic function, continuous on the entire number line and not identically constant, has a smallest positive period. Give an example of a periodic function, defined on the entire number line and not identically constant, that has no smallest positive period.

?

Problem 1.10.122

Let $f$ and $g$ be continuous functions, periodic with period $T$ , and $\lim_{x \rightarrow \infty }(f(x)-g(x)) = 0$ . Prove that $f = g$ .

?

Problem 1.10.123

A continuous function $f$ has two incommensurable periods, $T_{1}$ and $T_{2}$ (i.e., $T_{1} / T_{2}$ is an irrational number). Prove that $f =$ const.

?

Problem 1.10.124

Construct functions $f$ and $g$ , continuous on $\mathbb {R}$ , such that for any $T > 0$ the functions $f(x+T)$ and $g(x)$ are different, but for every $\varepsilon > 0$ there is $T_{\varepsilon } > 0$ such that for any $x \in \mathbb {R}$ the inequality holds

\left|f\left(x+T_{\varepsilon }\right)-g(x)\right| < \varepsilon

?

Problem 1.10.125

Let the function $f$ be continuous on $[-1 ; 1]$ and $\left|f(x)\right| < 1, x \in [-1 ; 1]$ . Prove that the function

f(x)-\cos (n \arccos x), \quad n \in \mathbb {N}

(see problem 227, §7), vanishes at least $n$ times on the closed interval $[-1 ; 1]$ .

?

Problem 1.10.126

Let $P(x) = 2^{n-1} x^{n}+a_{1} x^{n-1}+\ldots +a_{n}$ . Prove that there is $\xi \in [-1 ; 1]$ such that $\left|P(\xi )\right| \geqslant 1$ .

?

Problem 1.10.127

The function $f$ is continuous on $[0 ; 1]$ and $f(0) = f(1) = 0$ . Prove that there exists a continuous, concave (problem 237, §7) function $g$ such that $g(0) = g(1) = 0$ and $g(x) \geqslant f(x)$ on $[0 ; 1]$ .

?

Problem 1.10.128

The function $f$ is continuous and periodic with period $T$ . Prove that there is a point $x_{0}$ such that

f\left(x_{0}+T / 2\right) = f\left(x_{0}\right)

?

Problem 1.10.129

The function $f$ is continuous, monotone on $[0 ; 1]$ , and $f(0) = 0$ , $f(1) = 1$ . Prove that if for some $n \in \mathbb {N}$ , for any $x \in [0 ; 1]$ ,

\underbrace{f \circ f \circ \ldots \circ f}_{n \text{ times }}(x) = x,

then also $f(x) = x$ on $[0 ; 1]$ .

?

Problem 1.10.130

Let the function $f$ be continuous on $[0 ; 1]$ and let its range be contained in $[0 ; 1]$ . Prove that there exists a point $c \in [0 ; 1]$ such that $f(c) = c$ (every continuous map of a closed interval into itself has a fixed point).

?

Problem 1.10.131

Let $f$ and $g$ be defined and continuous on the closed interval $[a ; b]$ , with $f(a) < g(a)$ and $f(b) > g(b)$ . Prove that there is a point $c \in (a ; b)$ such that $f(c) = g(c)$ .

?

Problem 1.10.132

The functions $f$ and $g$ are defined and continuous on $[0 ; 1]$ , and $f \circ g = g \circ f$ . Prove that there exists a point $c \in [0 ; 1]$ such that $f(c) = g(c)$ .

?

Problem 1.10.133

Continuous functions $f$ and $g$ map the closed interval $[0 ; 1]$ onto itself. Prove that there exists a point $c \in [0 ; 1]$ such that $f(g(c)) = g(f(c))$ .

?

Problem 1.10.134

The function $f$ is continuous on $\mathbb {R}$ , and $f(f(x)) = x$ for any $x \in \mathbb {R}$ . Prove that there exists a point $c$ at which $f(c) = c$ .

?

Problem 1.10.135

(1)(2)

Give an example of a function:

?

(1)

continuous on the interval $(0 ; 1)$ ;

(2)

continuous on $\mathbb {R}$ , for which the equation $f(x) = x$ : has no solutions.

Problem 1.10.136

The function $f$ is monotone, continuous on $[0 ; 1]$ , and $0 \leqslant f(x) \leqslant 1$ for any $x \in [0 ; 1]$ . Prove that for any $a_{1} \in [0 ; 1]$ the sequence $a_{1}, a_{n+1} = f\left(a_{n}\right), n \in \mathbb {N}$ , converges to one of the solutions of the equation $f(x) = x$ .

?

Problem 1.10.137

(1)(2)

Let the function $f$ be continuous on $[a ; b]$ , and let the sequence $\left\{ x_{n}\right\} : x_{0} \in [a ; b], x_{n} = f\left(x_{n-1}\right), n \in \mathbb {N}$ (i.e., for any $n \in \mathbb {N} \quad f\left(x_{n-1} \in [a ; b]\right)$ ) be defined.

Prove that:

?

(1)

if $f$ is increasing, then $\left\{ x_{n}\right\}$ is a monotone sequence and there exists $\lim_{n \rightarrow \infty } x_{n} = c$ such that $c = f(c)$ ;

(2)

if $f$ is decreasing, then $\left\{ x_{2 k}\right\}$ and $\left\{ x_{2 k-1}\right\}$ are monotone subsequences having limits. Obtain equations for these limits.

Problem 1.10.138

(1)(2)

The function $f$ is strictly increasing, while the function $g$ is strictly decreasing on $[a ; b]$ , and $\mathbb {E}\left[f\right] \cap \mathbb {E}\left[g\right] \neq \varnothing$ .

?

(1)

Prove that the equation $f(x) = g(x)$ has a solution, and that it is unique.

(2)

Let $x_{0} \in [a ; b]$ be such that $f\left(x_{0}\right) \in \mathbb {E}\left[g\right]$ , and for any $n \in \mathbb {N}$ the equation $g\left(x_{n}\right) = f\left(x_{n-1}\right)$ has a solution $x_{n}$ , i.e., a sequence $\left\{ x_{n}\right\}$ is defined. Prove that the subsequences $\left\{ x_{2 k}\right\}$ and $\left\{ x_{2 k-1}\right\}$ converge, each to one of the solutions of the equation $g^{-1}(f(x)) = f^{-1}(g(x))$ .

Problem 1.10.139

For the sequence $\left\{ x_{n}\right\}$ , given recursively: $x_{1} = 1 / 2, x_{n+1} = f\left(x_{n}\right), n \in \mathbb {N}$ , where

f(x) = \begin{cases} (2-3 x) / 5, & x < -1 / 6 \\ -3 x, & \left|x\right| \leqslant 1 / 6 \\ -(2+3 x) / 5, & x > 1 / 6\end{cases}

find the limits of the subsequences $\left\{ x_{2 k}\right\}$ and $\left\{ x_{2 k-1}\right\}$ . Plot the graph of the function $f$ and show, in the figure, the construction of the first five terms of the sequence.

?

Problem 1.10.140

The function $f$ , defined on $\mathbb {R}$ , satisfies the Lipschitz condition: there exists $k > 0$ such that for any $x_{1}, x_{2} \in \mathbb {R}$ the inequality holds

\left|f\left(x_{1}\right)-f\left(x_{2}\right)\right| \leqslant k\left|x_{1}-x_{2}\right|.

Prove that if $k < 1$ , then there exists a solution of the equation $f(x) = x$ , and it is the only one.

?

Problem 1.10.141

(1)(2)

The range of the function $f$ , defined on $[a ; b]$ , is contained in $[a ; b]$ . For any $x, y \in [a ; b], x \neq y$ , the inequality holds

\left|f(x)-f(y)\right| < \left|x-y\right|

?

(1)

Prove that the equation $f(x) = x$ has a solution $c$ , and that it is the only one.

(2)

Let $x_{0} \in [a ; b], x_{n} = f\left(x_{n-1}\right), n \in \mathbb {N}$ . Prove that: a) the sequence $\left\{ \left|x_{n}-c\right|\right\}$ is decreasing and has a limit $\lim_{n \rightarrow \infty }\left|x_{n}-c\right| = \Delta$ б) there exists a subsequence $\left\{ x_{n_{k}}\right\}$ converging to $d$ , equal either to $c+\Delta$ or to $c-\Delta$ ; в) $\left|f(d)-c\right| = \Delta$ and $\Delta = 0$ , i.e., $\lim_{n \rightarrow \infty } x_{n} = c$ .

Problem 1.10.142

(1)(2)

?

(1)

Prove that the equation $\operatorname {tg} x = a / x, a > 0$ , has exactly one solution on each interval $(-\pi / 2+\pi n ; \pi / 2+\pi n), n \in \mathbb {N}$ .

(2)

Let $x_{n}$ be the solution of the equation $\operatorname {tg} x = a / x, a > 0$ , lying in the interval $(-\pi / 2+\pi n ; \pi / 2+\pi n), n \in \mathbb {N}$ . Prove that

0 < x_{n}-\pi n < \frac{2 a}{\pi n+\sqrt{\pi ^{2} n^{2}+4 a}}, \quad n \in \mathbb {N}

Problem 1.10.143

(1)(2)(3)

?

(1)

Prove that the equation $\operatorname {tg} x = a x, \quad a > 0$ , has exactly one solution $x_{n}$ on each interval $(-\pi / 2+\pi n ; \pi / 2+\pi n), n \in \mathbb {N}$ .

(2)

Let $x_{n}$ be the solution of the equation $\operatorname {tg} x = a x, a > 0$ , lying in the interval $(-\pi / 2+\pi n ; \pi / 2+\pi n), n \in \mathbb {N}$ . Prove that for all sufficiently large $n \in \mathbb {N}$

0 < \frac{\pi }{2}+\pi n-x_{n} < \frac{2 / a}{\pi / 2+\pi n+\sqrt{(\pi / 2+\pi n)^{2}}-4 / a}.

(3)

Find $\lim_{n \rightarrow \infty }\left(x_{n+1}-x_{n}\right)$ , where the sequence $\left\{ x_{n}\right\}$ is defined in 1) and 2).

Problem 1.10.144

For a sequence given recursively, prove that a limit exists and find it:

?

(1)

$x_{1} \in \mathbb {R}, x_{n+1} = \sin x_{n}, n \in \mathbb {N}$

(2)

$x_{1} = 0, x_{n+1} = x_{n}-\sin x_{n}+1 / 2, n \in \mathbb {N}$ ;

(3)

$x_{1} = \pi / 2, x_{n+1} = x_{n}+\cos x_{n}-1 / 2, n \in \mathbb {N}$ ;

(4)

$x_{1} \in \mathbb {R}, x_{n+1} = \operatorname {arctg} x_{n}, n \in \mathbb {N}$

(5)

$x_{1} = 0, x_{n+1} = x_{n}-\operatorname {arctg} x_{n}+\pi / 4, n \in \mathbb {N}$ ;

(6)

$x_{1} = 2, x_{n+1} = 1+\ln x_{n}, n \in \mathbb {N}$ .

Problem 1.10.145

(1)(2)

?

(1)

Let $\lim_{n \rightarrow \infty } x_{n} = \infty$ . Prove that $\lim_{n \rightarrow \infty }\left(1+\frac{1}{x_{n}}\right)^{x_{n}} = e$ .

(2)

Let $\lim_{n \rightarrow \infty } x_{n} = 0, x_{n} \neq 0$ . Prove that $\lim_{n \rightarrow \infty }\left(1+x_{n}\right)^{1 / x_{n}} = e$ .

Problem 1.10.146

Let $a_{1}, a_{2}, b_{1}, b_{2} \in \mathbb {R}$ . Investigate the sequence for convergence

\left\{ \left(\frac{a_{1} n+b_{1}}{a_{2} n+b_{2}}\right)^{n}\right\}

and find its limit, if it exists.

?

Problem 1.10.147

Using the continuity of the corresponding functions, compute the limit of the sequence:

?

(1)

$\left\{ \left(1+\frac{x_{n}}{n}\right)^{n}\right\}$ , if $\lim_{n \rightarrow \infty } x_{n} = x \in \mathbb {R}$ ;

(2)

$\left\{ \left(\cos \frac{x}{n}+\lambda \sin \frac{x}{n}\right)^{n}\right\}$ , where $x, \lambda \in \mathbb {R}$ ;

(3)

$\left\{ \left(\frac{\sqrt[n]{a}+\sqrt[n]{b}}{2}\right)^{n}\right\}$ , where $a > 0, b > 0$ ;

(4)

$\left\{ \left(\left(1+\frac{a}{n}\right)\left(1+\frac{2 a}{n}\right) \ldots \left(1+\frac{k a}{n}\right)\right)^{n}\right\}$ , where $k \in \mathbb {N}, a \in \mathbb {R}$ ;

(5)

$\left\{ \left(1+\frac{1}{n^{2}}\right)\left(1+\frac{2}{n^{2}}\right) \ldots \left(1+\frac{n}{n^{2}}\right)\right\}$ ;

(6)

$\left\{ \cos \frac{x}{2} \cos \frac{x}{2^{2}} \ldots \cos \frac{x}{2^{n}}\right\}$ ;

(7)

$\left\{ \sin^{2}\left(\pi \sqrt{n^{2}+n}\right)\right\}$ ;

(8)

$\left\{ n-\frac{1}{\sin (1 / n)}\right\}$ ;

(9)

$\left\{ n-\operatorname {ctg} \frac{1}{n}\right\}$ ;

(10)

$\left\{ (\cos (x / \sqrt{n}))^{n}\right\}$ .

Problem 1.10.148

The sequences $\left\{ a_{n}\right\}$ and $\left\{ b_{n}\right\}$ are such that $0 < a_{n} < 1$ , $\lim_{n \rightarrow \infty } a_{n} = 1,0 < b_{n} < \frac{\pi }{2}, \cos b_{n} = a_{n}$ . Find $\lim_{n \rightarrow \infty } \frac{b_{n}}{\sqrt{1-a_{n}}}$ .

?