Math Problems Vault

Problem 1.12.1

Prove that the function $f$ is uniformly continuous on the set $X$ , if:

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(1)

$f(x) = 2 x-1, X = R$

(2)

$f(x) = x^{2}, X = (-1 ; 1)$

(3)

$f(x) = \sqrt[3]{x}, X = [0 ; 2]$

(4)

$f(x) = \sin x^{2}, X = (-3 ; 3]$ ;

(5)

$f(x) = x \sin (1 / x), X = (0 ; \pi ]$ .

Problem 1.12.2

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

Prove that the function is not uniformly continuous on the set $X$ :

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(1)

$y = \cos (1 / x), X = (0 ; 1)$ ;

(2)

$y = \sin x^{2}, x = R$

(3)

$y = x^{2}, X = R$ ;

(4)

$y = \ln x, X = (0 ; 1)$ .

Problem 1.12.3

Investigate the function for uniform continuity on the set $X$ .

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(1)

$y = e^{-\arcsin x}, X = [-1 ; 1]$ ;

(2)

$y = \operatorname {arctg}\left(\frac{\ln (1+x)}{\sqrt{x^{2}+1}+\left|\sin x\right|}\right), X = [0 ; 10]$ ;

(3)

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

(4)

$y = e^{x}, X = R$ ;

(5)

$y = \operatorname {ctg} x, X = (0 ; 1)$ ;

(6)

$y = \frac{x^{6}-1}{\sqrt{1-x^{4}}}, X = (-1 ; 1)$ ;

(7)

$y = \sin \sqrt{x}, X = [1 ;+\infty )$ ;

(8)

$y = \begin{cases} 1-x^{2}, & -1 \leqslant x \leqslant 0, \\ 1+x, & 0 < x \leqslant 1,\end{cases} \quad X = [-1 ; 1]$

(9)

$y = x \sin (1 / x), X = (0 ;+\infty )$ .

Problem 1.12.4

Investigate the function for uniform continuity on the set $X$ .

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(1)

$y = \begin{cases} x+1, & x \leqslant 0, \quad X = R \\ e^{-x}, & x > 0, \end{cases}$

(2)

$y = \cos x \cos \frac{\pi }{x}, X = (0 ; 1)$ ;

(3)

$y = \frac{\sin x}{x}, X = (0 ; \pi )$ ;

(4)

$y = x+\sin x, X = R$ ;

(5)

$y = x \cos x, X = R$ ;

(6)

$y = \sin (1 / x), X = [0,01 ;+\infty )$ ;

(7)

$y = n^{2}$ for $2 n \leqslant x \leqslant 2 n+1, n \in \mathbb {N}, X$ is the union of all closed intervals $[2 n ; 2 n+1], n \in \mathbb {N}$ ;

(8)

$y = \frac{\left|\sin x\right|}{x}, X = (-\pi ; 0) \cup (0 ; \pi )$ .

Problem 1.12.5

The function $f$ satisfies the following condition on the set $X$ : there exist numbers $k > 0$ and $\alpha > 0$ such that for any $x_{1}$ and $x_{2}$ in $X$ the inequality holds

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

(for $\alpha = 1$ this condition is called the Lipschitz condition; for $\alpha < 1$ , the Hölder condition of order $\alpha$ ). Prove that a function satisfying this condition is uniformly continuous on the set $X$ .

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Problem 1.12.6

Prove that if a function is uniformly continuous on an interval, then it is also continuous on this interval.

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Problem 1.12.7

Prove that if a function is unbounded on a bounded interval, then it is not uniformly continuous on this interval.

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Problem 1.12.8

(1)(2)

Give an example of a function:

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(1)

bounded and continuous on a bounded interval, but not uniformly continuous on it;

(2)

continuous on a closed (see problem 94, §10) set and not uniformly continuous on it.

Problem 1.12.9

(1)(2)

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(1)

Prove that if the function $f$ is uniformly continuous on a bounded set, then it is bounded on this set.

(2)

Give an example of a function uniformly continuous on a set and unbounded on this set.

Problem 1.12.10

Draw the graph of the modulus of continuity $\omega (\delta )$ of the function $f$ on the set $X$ , if:

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(1)

$f(x) = 1-2 x, X = R$ ;

(2)

$f(x) = \left|x\right|, X = R$ ;

(3)

$f(x) = x, X = [-2 ;-1] \cup [1 ; 2]$

(4)

$f(x) = \frac{1}{2 a}(\left|x+a\right|-\left|x-a\right|), a > 0, X = R$ ;

(5)

$f(x) = x^{3}, X = [0 ; 1]$ ;

(6)

$f(x) = \mathbb {E}\left[x\right], x \in \mathbb {R}$ .

Problem 1.12.11

Find the modulus of continuity of the function $f$ on the set $X$ and, using it, establish the uniform continuity of the function, if:

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(1)

$f(x) = 2 x-1, X = R$ ;

(2)

$f(x) = x^{2}, X = [-a ; a], a > 0$ ;

(3)

$f(x) = 1 / x, X = [a ;+\infty ), a > 0$ ;

(4)

$f(x) = \cos x, X = R$ ;

(5)

$f(x) = \ln x, X = [1 ;+\infty )$ .

Problem 1.12.12

(1)(2)

Prove that for any $\delta > 0$ , $\omega (\delta ; f ; X) = +\infty$ , if:

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(1)

$f(x) = x^{3}, X = R$ ;

(2)

$f(x) = 1 / x^{2}, X = (0 ; 1)$ .

Problem 1.12.13

(1)(2)

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(1)

Give an example of a function whose modulus of continuity $\omega (\delta )$ satisfies the condition $\omega (+0) = \varepsilon > 0$ , where $\varepsilon$ is a given number.

(2)

Give an example of a function unbounded on an interval, whose modulus of continuity $\omega (\delta )$ on this interval satisfies the condition $\omega (\delta ) < +\infty$ for any $\delta > 0$ .

Problem 1.12.14

Let $\omega (\delta )$ be the modulus of continuity of the function $f$ on the set $X$ .

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(1)

Prove that if $\delta_{1} < \delta_{2}$ , then $\omega \left(\delta_{1}\right) \leqslant \omega \left(\delta_{2}\right)$ .

(2)

Prove that if $f$ is bounded on $X$ , then $\omega (\delta ) < +\infty$ for any $\delta > 0$ .

(3)

Prove that if the set $X$ is bounded, but the function $f$ is unbounded on $X$ , then $\omega (\delta ) = +\infty$ for any $\delta > 0$ .

(4)

Prove that if $\omega \left(\delta_{0}\right)$ is a (finite) number for some $\delta_{0}$ , then for every $\delta , 0 < \delta < \delta_{0}, \omega (\delta )$ is also a number, i.e., $\omega (\delta )$ is a function on $\left(0 ; \delta_{0}\right)$ , and there exists

\lim _{\delta \rightarrow +0} \omega (\delta ) = \omega (+0)

(5)

Prove that if $X$ is an interval, then for any $\delta_{1} > 0$ , $\delta_{2} > 0$

\omega \left(\delta _{1}+\delta _{2}\right) \leqslant \omega \left(\delta _{1}\right)+\omega \left(\delta _{2}\right)

(6)

Prove that if $X$ is an interval and $\omega \left(\delta_{0}\right) = +\infty$ for some $\delta_{0}$ , then $\omega (\delta ) = +\infty$ for any $\delta > 0$ .

(7)

Give an example of a function for which $\omega (\delta ) = \delta$ for $0 < \delta < 1$ ; $\omega (\delta ) = +\infty$ for $\delta \geqslant 1$ .

Problem 1.12.15

Prove that for any $\varepsilon > 0$ there exists $\delta$ such that for any $x_{1}, x_{2} \in D(f)$ the inequality $\left|x_{1}-x_{2}\right| < \delta$ implies the inequality

\left|f\left(x_{1}\right)-f\left(x_{2}\right)\right| < \omega (+0)+\varepsilon .

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Problem 1.12.16

Prove that if the function $f$ is uniformly continuous on the set $X$ , and $Y \in X$ , then $f$ is also uniformly continuous on the set $Y$ .

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Problem 1.12.17

The function $f$ is uniformly continuous on the closed intervals $[a ; b]$ and $[b ; c]$ . Prove that it is also uniformly continuous on the closed interval $[a ; c]$ .

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Problem 1.12.18

Give an example of a function uniformly continuous on the closed interval $[a ; b]$ and on the half-open interval $(b ; c]$ , but not uniformly continuous on the closed interval $[a ; c]$ .

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Problem 1.12.19

Prove that if the functions $f$ and $g$ are uniformly continuous on the set $X$ , then for any $\alpha , \beta \in \mathbb {R}$ the function $\alpha f+\beta g$ is uniformly continuous on $X$ .

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Problem 1.12.20

(1)(2)(3)

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(1)

Prove that if the functions $f$ and $g$ are bounded and uniformly continuous on $[a ;+\infty )$ , then their product $f g$ is also a function uniformly continuous on $[a ;+\infty )$ .

(2)

Give an example of functions, uniformly continuous on $[a ;+\infty )$ , whose product is not a function uniformly continuous on $[a ;+\infty )$ .

(3)

Prove that if the functions $f$ and $g$ are uniformly continuous on a bounded set, then their product $f g$ is a function uniformly continuous on this set.

Problem 1.12.21

Prove that a continuous periodic function is uniformly continuous.

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Problem 1.12.22

The function $f$ is uniformly continuous on $[a ;+\infty )$ . Prove that one of three conditions holds: either $f(x)$ is bounded on $[a ;+\infty )$ , or $\lim_{x \rightarrow +\infty } f(x) = +\infty$ , or $\lim_{x \rightarrow +\infty } f(x) = -\infty$ .

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Problem 1.12.23

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

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Problem 1.12.24

Prove that a function bounded, monotone, and continuous on an interval is uniformly continuous on this interval (finite or infinite).

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Problem 1.12.25

Prove that for the uniform continuity of the function $f$ on a bounded interval $(a ; b)$ , it is necessary and sufficient that the function be continuous on $(a ; b)$ and that the limits $\lim_{x \rightarrow a+0} f(x)$ and $\lim_{x \rightarrow b-0} f(x)$ exist.

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Problem 1.12.26

Prove that for a function $f$ , defined and continuous on the interval $(a ; b)$ , to be extendable as a continuous function to the closed interval $[a ; b]$ , it is necessary and sufficient that the function $f$ be uniformly continuous on the interval $(a ; b)$ .

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Problem 1.12.27

Prove that if the function $f$ is uniformly continuous on the interval $(a ; b)$ , then it can be extended as a continuous function to the entire number line $\mathbb {R}$ , i.e., there exists a continuous function $F(x)$ , defined on $\mathbb {R}$ , such that $F(x) = f(x)$ for any $x \in (a ; b)$ .

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Problem 1.12.28

The function $f$ is uniformly continuous on $\mathbb {R}$ . Prove that there exist numbers $a \geqslant 0, b \geqslant 0$ such that $\left|f(x)\right| \leqslant a\left|x\right|+b$ for any $x \in \mathbb {R}$ .

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Problem 1.12.29

A function $f$ , continuous on the closed interval $[a ; b]$ , is called piecewise linear if there exists a partition of the closed interval $[a ; b]$ by points $x_{0}, x_{1}, \ldots , x_{n}$ such that

a = x_{0} < x_{1} < \ldots < x_{i} < \ldots < x_{n-1} < x_{n} = b

and the function $f$ is linear on each closed interval $\left[x_{i-1} ; x_{i}\right], i = 1,2, \ldots , n$ (i.e., $f(\mathbf{x}) = a_{i} x+b_{i}$ ). Prove that every function $F$ , continuous on the closed interval $[a ; b]$ , can be approximated to any degree of accuracy by a piecewise linear function, i.e., for any $\varepsilon > 0$ there exists a piecewise linear function $f$ such that for all $x \in [a ; b]$ the inequality holds

\left|f(x)-F(x)\right| < \varepsilon

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Problem 1.12.30

(1)(2)

For the function $F$ , choose a piecewise linear function $f$ so that for all $x \in [a ; b]$ the inequality $\left|F(x)-f(x)\right| < 0,1$ holds:

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(1)

$F(x) = x^{2},[a ; b] = [-1 ; 1]$

(2)

$F(x) = 1 / x,[a ; b] = [2 / 3 ; 2]$ .

Problem 1.12.31

Find a piecewise linear function $f$ on the closed interval $[0 ; 100]$ such that for all $x \in [0 ; 100]$ the inequality holds

\left|2^{-x}-f(x)\right| < 1 / 4

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Problem 1.12.32

A function $f$ , continuous on $\mathbb {R}$ , is called piecewise linear on $\mathbb {R}$ if there exist numbers $a$ and $b, a < b$ , such that the function $f$ is linear on the intervals $(-\infty ; a],[b ;+\infty )$ and piecewise linear on $[a ; b]$ . Prove that, whatever the number $\varepsilon$ may be, there is no function $f$ , piecewise linear on $\mathbb {R}$ , for which $\left|x^{2}-f(x)\right| < 1$ for all $x \in \mathbb {R}$ .

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Problem 1.12.33

Prove that every piecewise linear function can be given by the formula

f(x) = a+\sum _{i = 0}^{n} a_{i}\left|x-x_{i}\right|

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Problem 1.12.34

(1)(2)(3)

Find the formula indicated in the preceding problem for a function $f$ , piecewise linear on a closed interval, if:

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(1)

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

(2)

$f(x) = \begin{cases} 2 x+3, & -2 \leqslant x \leqslant 0 \\ 3-x, & 0 < x \leqslant 1 \\ x+1, & 1 < x \leqslant 3 \\ 4, & 3 \leqslant x \leqslant 5\end{cases}$

(3)

$f(0) = 1, f(1) = -1, f(3) = 2, f(4) = -4, f(6) = 0$ , where $\left\{ 0,1\text{, }3,4,6\right\}$ is a partition of the closed interval $[0 ; 6]$ .

Problem 1.12.35

Prove that a function piecewise linear on $\mathbb {R}$ is uniformly continuous on $\mathbb {R}$ .

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Problem 1.12.36

Let

f(x) = \begin{cases} \sin (1 / x), & x \neq 0, \quad \left|x\right| \leqslant 1 / \pi \\ a, & x = 0\end{cases}

Prove that for no value of $a$ does there exist a function $g$ , piecewise linear on $[-1 / \pi ; 1 / \pi ]$ , such that $\left|f(x)-g(x)\right| < 1 / 2$ for any $x \in [-1 / \pi ; 1 / \pi ]$ .

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Problem 1.12.37

Formulate in positive form the statement that a function, defined on a closed interval, cannot be approximated to any degree of accuracy by a piecewise linear function.

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Problem 1.12.38

Prove that if a function $f$ is defined, but discontinuous, on the closed interval $[a ; b]$ , then it cannot be approximated to any degree of accuracy by a piecewise linear function.

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Problem 1.12.39

Prove that if a function $f$ on an interval can be approximated to any degree of accuracy by a piecewise linear function, then $f$ is uniformly continuous on this interval.

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Problem 1.12.40

Prove that if the function $f$ on the interval $[c ;+\infty )$ can be approximated to any degree of accuracy by a piecewise linear function, then $f$ has an asymptote as $x \rightarrow +\infty$ (a function is piecewise linear on $[c ;+\infty )$ if it is continuous on $[c ;+\infty )$ , linear on $[d ;+\infty )$ for some $d > c$ , and piecewise linear on $[c ; d]$ ).

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