Continuity of Functions

The graph of a continuous function $y=f(x)$ is given. For a given point $a$ and a given number $\varepsilon >0$, indicate geometrically a number $\delta >0$ such that when $\left|x-a\right|<\delta$, $\left|f(x)-f(a)\right|<\varepsilon$.

A metal square plate with side $x_{0}=10$ cm is to be made. If the difference between its area $y=x^{2}$ and the intended $y_{0}=100$ square cm must not exceed (1) $\pm 1$ square cm; (2) $\pm 0.1$ square cm; (3) $\pm 0.01$ square cm; (4) $\pm \varepsilon$ square cm, within what range may the side $x$ vary?

The edge of a cube lies between 2 meters and 3 meters. In order that the absolute error incurred in computing the volume of this cube should not exceed $\varepsilon$ cubic meters, letting (1) $\varepsilon =0.1$ cubic meters; (2) $\varepsilon =0.01$ cubic meters; (3) $\varepsilon =0.001$ cubic meters, what absolute error $\triangle$ is allowed when measuring the edge $x$ of this cube?

In what largest possible neighborhood of $x_{0}=100$ does the difference between the ordinate of the graph of the function $y=\sqrt{x}$ and $y_{0}=10$ remain less than $\varepsilon =10^{-n}$ ($n\geqslant 0$)? Find the size of this neighborhood for $n=0,1,2,3$.

Using the $\varepsilon$-$\delta$ argument, prove that the function $f(x)=x^{2}$ is continuous at $x=5$. Fill in the following table:

Let $f(x)=\dfrac {1}{x}$ and $\varepsilon =0.001$. For the values $x_{0}=0.1;0.01;0.001;\ldots$ find a sufficiently large positive number $\delta =\delta (\varepsilon ,x_{0})$ such that the inequality $\left|x-x_{0}\right|<\delta$ implies the inequality

Can one, for the given $\varepsilon =0.001$, choose a $\delta >0$ that works for all values $x_{0}$ in the interval $(0,1)$, in other words, such that for any value $x_{0}\in (0,1)$, if $\left|x-x_{0}\right|<\delta$, then $\left|f(x)-f(x_{0})\right|<\varepsilon$?

Express concisely, in positive form, using the $\varepsilon$-$\delta$ language, the following assertion: the function $f(x)$ is defined at the point $x_{0}$, but is not continuous there.

Suppose that for certain numbers $\varepsilon >0$ one can find a corresponding number $\delta =\delta (\varepsilon ,x_{0})>0$ such that, whenever $\left|x-x_{0}\right|<\delta$, then

Suppose: (1) the numbers $\varepsilon$ form a finite set; (2) the numbers $\varepsilon$ form the infinite set of fractions $\varepsilon =\dfrac {1}{2^{n}}$ ($n=1,2,\ldots$). Can one conclude that the function $f(x)$ is continuous at the point $x_{0}$?

Given the function

Prove that for every $\varepsilon >0.001$, one can choose $\delta =\delta (\varepsilon ,x)>0$ such that: whenever $\left|x'-x\right|<\delta$, then $\left|f(x')-f(x)\right|<\varepsilon$. But for $0<\varepsilon \leqslant 0.001$, this is not possible for any value of $x$.

At which points does this function lose continuity?

Suppose that for every sufficiently small number $\delta >0$ there is $\varepsilon =\varepsilon (\delta ,x_{0})>0$ such that: whenever $\left|x-x_{0}\right|<\delta$, the inequality $\left|f(x)-f(x_{0})\right|<\varepsilon$ holds. Can one conclude from this that the function $f(x)$ is continuous at $x=x_{0}$? What property of the function $f(x)$ does this known inequality express?

Suppose that for every number $\varepsilon >0$ there is a number $\delta =\delta (\varepsilon ,x_{0})>0$ such that: if $\left|f(x)-f(x_{0})\right|<\varepsilon$, then $\left|x-x_{0}\right|<\delta$.

Can one conclude from this that the function $f(x)$ is continuous at $x=x_{0}$? What property of the function do these inequalities express?

Suppose that for every number $\delta >0$ and every $x=x_{0}$, there is a number $\varepsilon =\varepsilon (\delta ,x_{0})>0$ such that: if $\left|f(x)-f(x_{0})\right|<\varepsilon$, then $\left|x-x_{0}\right|<\delta$.

Can one conclude from this that the function $f(x)$ is continuous at $x=x_{0}$? What property of the function does this known inequality express?

Using the $\varepsilon$-$\delta$ argument, prove the continuity of the following functions:

Investigate the continuity of the following function and sketch its graph: $f(x)=\left|x\right|$.

Investigate the continuity of the following function and sketch its graph:

Investigate the continuity of the following function and sketch its graph: if $x\neq -1$, $f(x)=\dfrac {1}{(1+x)^{2}}$, and $f(-1)$ is arbitrary.

Investigate the continuity of the following functions and sketch their graphs:

Investigate the continuity of the following function and sketch its graph: if $x\neq 0$, $f(x)=\sin \dfrac {1}{x}$, and $f(0)$ is arbitrary.

Investigate the continuity of the following function and sketch its graph: if $x\neq 0$, $f(x)=x\sin \dfrac {1}{x}$, and $f(0)=0$.

Investigate the continuity of the following function and sketch its graph: if $x\neq 0$, $f(x)=e^{-\frac{1}{x^{2}}}$, and $f(0)=0$.

Investigate the continuity of the following function and sketch its graph: if $x\neq 1$, $f(x)=\dfrac {1}{1+e^{\frac{1}{x-1}}}$, and $f(1)$ is arbitrary.

Investigate the continuity of the following function and sketch its graph: if $x\neq 0$, $f(x)=x\ln x^{2}$, and $f(0)=a$.

Investigate the continuity of the following function and sketch its graph: $f(x)=\operatorname {sgn}x$.

Investigate the continuity of the following function and sketch its graph: $f(x)=\left[x\right]$.

Investigate the continuity of the following function and sketch its graph: $f(x)=\sqrt{x}-\left[\sqrt{x}\right]$.

Find the discontinuities of the following function and investigate the character of these points: $y=\dfrac {x}{(1+x)^{2}}$.

Find the discontinuities of the following function and investigate the character of these points: $y=\dfrac {1+x}{1+x^{3}}$.

Find the discontinuities of the following function and investigate the character of these points: $y=\dfrac {x^{2}-1}{x^{3}-3x+2}$.

Find the discontinuities of the following function and investigate the character of these points: $y=\dfrac {\dfrac {1}{x}-\dfrac {1}{x+1}}{\dfrac {1}{x-1}-\dfrac {1}{x}}$.

Find the discontinuities of the following function and investigate the character of these points: $y=\dfrac {x}{\sin x}$.

Find the discontinuities of the following function and investigate the character of these points: $y=\sqrt{\dfrac {1-\cos \pi x}{4-x^{2}}}$.

Find the discontinuities of the following function and investigate the character of these points: $y=\cos^{2}\dfrac {1}{x}$.

Find the discontinuities of the following function and investigate the character of these points: $y=\operatorname {sgn}\left(\sin \dfrac {\pi }{x}\right)$.

Find the discontinuities of the following function and investigate the character of these points: $y=\dfrac {\cos \dfrac {\pi }{x}}{\cos \dfrac {\pi }{x}}$.

Find the discontinuities of the following function and investigate the character of these points: $y=\operatorname {arctg}\dfrac {1}{x}$.

Find the discontinuities of the following function and investigate the character of these points: $y=\sqrt{x}\operatorname {arctg}\dfrac {1}{x}$.

Find the discontinuities of the following function and investigate the character of these points: $y=e^{x+\frac{1}{x}}$.

Find the discontinuities of the following function and investigate the character of these points: $y=\dfrac {1}{\ln x}$.

Find the discontinuities of the following function and investigate the character of these points: $y=\dfrac {1}{1-e^{\frac{x}{1-x}}}$.

Investigate the continuity of the following function and sketch its approximate graph: $y=\operatorname {sgn}(\sin x)$.

Investigate the continuity of the following function and sketch its approximate graph: $y=x-\left[x\right]$.

Investigate the continuity of the following function and sketch its approximate graph: $y=x\left[x\right]$.

Investigate the continuity of the following function and sketch its approximate graph: $y=\left[x\right]\sin \pi x$.

Investigate the continuity of the following function and sketch its approximate graph: $y=x^{2}-\left[x^{2}\right]$.

Investigate the continuity of the following function and sketch its approximate graph: $y=\left[\dfrac {1}{x}\right]$.

Investigate the continuity of the following function and sketch its approximate graph: $y=x\left[\dfrac {1}{x}\right]$.

Investigate the continuity of the following function and sketch its approximate graph: $y=\operatorname {sgn}\left(\cos \dfrac {1}{x}\right)$.

Investigate the continuity of the following function and sketch its approximate graph: $y=\left[\dfrac {1}{x^{2}}\right]\operatorname {sgn}\left(\sin \dfrac {\pi }{x}\right)$.

Investigate the continuity of the following function and sketch its approximate graph: $y=\operatorname {ctg}\dfrac {\pi }{x}$.

Investigate the continuity of the following function and sketch its approximate graph: $y=\sec^{2}\dfrac {1}{x}$.

Investigate the continuity of the following function and sketch its approximate graph: $y=(-1)^{\left[x^{2}\right]}$.

Investigate the continuity of the following function and sketch its approximate graph: $y=\operatorname {arctg}\left(\dfrac {1}{x}+\dfrac {1}{x-1}+\dfrac {1}{x-2}\right)$.

Investigate the continuity of the following function and sketch its approximate graph: $y=\dfrac {1}{x^{2}\sin^{2}x}$.

Investigate the continuity of the following function and sketch its approximate graph: $y=\dfrac {1}{\sin (x^{2})}$.

Investigate the continuity of the following function and sketch its approximate graph: $y=\ln \dfrac {x^{2}}{(x+1)(x-3)}$.

Investigate the continuity of the following function and sketch its approximate graph: $y=e^{-\frac{1}{x}}$.

Investigate the continuity of the following function and sketch its approximate graph: $y=1-e^{-\frac{1}{x^{2}}}$.

Investigate the continuity of the following function and sketch its approximate graph: $y=\operatorname {th}\dfrac {2x}{1-x^{2}}$.

Investigate the continuity of the following function and construct its graph: $y=\lim \limits_{n\to \infty }\dfrac {1}{1+x^{n}}$ ($x\geqslant 0$).

Investigate the continuity of the following function and construct its graph: $y=\lim \limits_{n\to \infty }\dfrac {n^{x}-n^{-x}}{n^{x}+n^{-x}}$.

Investigate the continuity of the following function and construct its graph: $y=\lim \limits_{n\to \infty }\sqrt[2n]{1+x^{2n}}$.

Investigate the continuity of the following function and construct its graph: $y=\lim \limits_{n\to \infty }\cos^{2n}x$.

Investigate the continuity of the following function and construct its graph: $y=\lim \limits_{n\to \infty }\dfrac {x}{1+(2\sin x)^{2n}}$.

Investigate the continuity of the following function and construct its graph: $y=\lim \limits_{n\to \infty }\left[x\operatorname {arctg}(n\operatorname {ctg}x)\right]$.

Investigate the continuity of the following function and sketch its graph: $y=\lim \limits_{n\to \infty }\dfrac {x+x^{2}e^{nx}}{1+e^{nx}}$.

Investigate the continuity of the following function and sketch its graph: $y=\lim \limits_{t\to +\infty }\dfrac {\ln (1+e^{tx})}{\ln (1+e^{t})}$.

Investigate the continuity of the following function and sketch its graph: $y=\lim \limits_{t\to +\infty }(1+x)\operatorname {th}(tx)$.

The function

is it a continuous function?

Let:

How should the number $a$ be chosen so that the function $f(x)$ is continuous?

Investigate the continuity of the following functions and describe the nature of the discontinuity points. Let:

The function $d=d(x)$ is the shortest distance between the point $x$ on the number axis $Ox$ and the point set formed by the segments $0\leqslant x\leqslant 1$ and $2\leqslant x\leqslant 3$. Find the explicit formula for the function $d$, sketch its graph, and investigate its continuity.

The figure $E$ is formed by an isosceles triangle with height 1 and base 1, together with two rectangles of base 1 and heights 2 and 3 (Figure 1.307). The function $S=S(y)$ ($0\leqslant y<+\infty$) is the area of the part of the figure $E$ lying between the lines $Y=0$ and $Y=y$; and the function $b=b(y)$ ($0\leqslant y<+\infty$) is the length of the cross-section cut from the figure by the line $Y=y$. Find the explicit formulas for the functions $S$ and $b$, sketch their graphs, and investigate their continuity.

Prove that the Dirichlet function

is discontinuous for every value of $x$.

Let

where $\chi (x)$ is the Dirichlet function (same as in the previous problem 734). Investigate the continuity of this function $f(x)$. Sketch this function.

Prove that the Riemann function

is discontinuous for every rational value of $x$, and continuous for every irrational value of $x$. Sketch this function.

Let $f(x)=\dfrac {nx}{n+1}$ when $x$ is the reduced rational fraction $\dfrac {m}{n}$ ($n\geqslant 1$); and $f(x)=\left|x\right|$ when $x$ is irrational. Investigate the continuity of the function $f(x)$ and sketch this function.

The function $f(x)=\dfrac {1-\cos x}{x^{2}}$ is defined for all values of the independent variable $x$ except $x=0$. In order for this function to be continuous at $x=0$, what value should be assigned to the function at the point $x=0$?

Prove that no matter how the number $f(1)$ is chosen, the function $f(x)=\dfrac {1}{1-x}$ is discontinuous at $x=1$.

At $x=0$, the function $f(x)$ is undefined; define the value of $f(0)$ so that $f(x)$ is continuous at the point $x=0$, if:

Let: (1) the function $f(x)$ be continuous at $x=x_{0}$, while the function $g(x)$ is discontinuous at $x=x_{0}$; (2) at $x=x_{0}$ both functions $f(x)$ and $g(x)$ are discontinuous. Must the sum $f(x)+g(x)$ of these two functions necessarily be discontinuous at the given point $x_{0}$? Give appropriate examples.

Let: (1) the function $f(x)$ be continuous at the point $x_{0}$, while $g(x)$ is discontinuous at the point $x_{0}$; (2) at $x=x_{0}$ both $f(x)$ and $g(x)$ are discontinuous. Must the product $f(x)g(x)$ of these two functions necessarily be discontinuous at the given point $x_{0}$? Give suitable examples.

Can it be asserted that the square of a discontinuous function is still discontinuous? Give an example of a function which is discontinuous everywhere, but whose square is a continuous function.

Investigate the continuity of the functions $f[g(x)]$ and $g[f(x)]$, given:

Let

and

Investigate the continuity of the composite function $y=f(u)$, where $u=\varphi (x)$.

Prove that if $f(x)$ is a continuous function, then the following function is also continuous:

Prove that if the function $f(x)$ is continuous, then the function

(where $c$ is an arbitrary positive number) is also a continuous function.

Prove that if the function $f(x)$ is continuous on the closed interval $\left[a,b\right]$, then the functions

are also continuous on $\left[a,b\right]$.

Prove that if the functions $f(x)$ and $g(x)$ are continuous, then the functions

are also continuous.

Let the function $f(x)$ be defined and bounded on the closed interval $\left[a,b\right]$. Prove that the functions

are left-continuous on the closed interval $\left[a,b\right]$. And the functions

are right-continuous on the closed interval \left[a,b\right]$$^{*)}.

Prove that if the function $f(x)$ is continuous on the interval $a\leqslant x<+\infty$, and $\lim \limits_{x\to +\infty }f(x)$ is finite, then this function is bounded on the given interval.

Let the function $f(x)$ be continuous and bounded on the interval $(x_{0},+\infty )$. Prove that for any number $T$, one can find a sequence $x_{n}\to +\infty$ such that

If $\varphi (x)$ and $\phi (x)$ are continuous periodic functions defined for $-\infty <x<+\infty$, and

prove that $\varphi (x)\equiv \phi (x)$.

Prove that every discontinuity of a monotone bounded function is a discontinuity of the first kind.

Prove that if the function $f(x)$ has the following properties: (1) it is defined and monotone on the closed interval $\left[a,b\right]$, (2) it takes as function values all numbers between $f(a)$ and $f(b)$, then this function is continuous on $\left[a,b\right]$.

Prove that the function

takes on any closed interval $\left[a,b\right]$ every intermediate value between $f(a)$ and $f(b)$, but is not continuous on $\left[a,b\right]$.

Prove that if the function $f(x)$ is continuous on the interval $(a,b)$, and $x_{1},x_{2},\ldots ,x_{n}$ are arbitrary values in this interval, then one can find a value $\xi$ between them such that

Let the function $f(x)$ be continuous on the interval $(a,b)$, and let

Prove that for any number $\lambda$, with $l\leqslant \lambda \leqslant L$, there exists a sequence $x_{n}\to a+0$ ($n=1,2,\ldots$) such that