Math Problems Vault

Problem 785

(a)(b)(c)

A certain factory's workshop manufactures square sheet-metal plates whose side $x$ can take any value from 1 centimeter to 10 centimeters. In order that, for whatever side length (within the range above), the area $y$ of the plate should differ from the intended design area by less than $\varepsilon$ , what tolerance $\delta$ may be used in machining the side length of these plates, given:

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(a)

$\varepsilon =1$ square centimeter;

(b)

$\varepsilon =0.01$ square centimeter;

(c)

$\varepsilon =0.0001$ square centimeter, compute the value of $\delta$ .

Problem 786

(a)(b)(c)(d)

A cylindrical sleeve of width $\varepsilon$ and length $\delta$ is fitted over the curve $y=\sqrt[3]{x}$ and slides along this curve, with the axis of the sleeve kept parallel to the $Ox$ axis. In order for this sleeve to pass smoothly over the portion of the curve bounded by the inequality $-10\leqslant x\leqslant 10$ , what should $\delta$ equal, given:

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(a)

$\varepsilon =1$ ;

(b)

$\varepsilon =0.1$ ;

(c)

$\varepsilon =0.001$ ;

(d)

$\varepsilon$ an arbitrary small number.

Problem 787

Using the `` $\varepsilon -\delta$ '' language, state affirmatively the meaning of the following assertion: the function $f(x)$ is continuous on a certain set (interval, segment), but is not uniformly continuous on this set.

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Problem 788

Prove that the function

f(x)=\dfrac {1}{x}

is continuous on the interval $(0,1)$ , but is not uniformly continuous on this interval.

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Problem 789

Prove that the function

f(x)=\sin \dfrac {\pi }{x}

is continuous and bounded on the interval $(0,1)$ , but is not uniformly continuous on this interval.

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Problem 790

Prove that the function

f(x)=\sin x^{2}

is continuous and bounded on the infinite interval $-\infty <x<+\infty$ , but is not uniformly continuous on this interval.

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Problem 791

Prove that if the function $f(x)$ is defined and continuous on the domain $a\leqslant x<+\infty$ , and if

\lim _{x\to +\infty }f(x)

exists, then $f(x)$ is uniformly continuous on this domain.

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Problem 792

Prove that the unbounded function

f(x)=x+\sin x

is uniformly continuous on the whole line $-\infty <x<+\infty$ .

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Problem 793

(a)(b)

Is the function $f(x)=x^{2}$ uniformly continuous on the following intervals:

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(a)

$(-l,l)$ , where $l$ is an arbitrarily large positive number;

(b)

on the interval $(-\infty ,+\infty )$ ?

Problem 794

Investigate the uniform continuity of the function $f(x)=\dfrac {x}{4-x^{2}}$ ( $-1\leqslant x\leqslant 1$ ) on the given domain.

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Problem 795

Investigate the uniform continuity of the function $f(x)=\ln x$ ( $0<x<1$ ) on the given domain.

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Problem 796

Investigate the uniform continuity of the function $f(x)=\dfrac {\sin x}{x}$ ( $0<x<\pi$ ) on the given domain.

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Problem 797

Investigate the uniform continuity of the function $f(x)=e^{x}\cos \dfrac {1}{x}$ ( $0<x<1$ ) on the given domain.

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Problem 798

Investigate the uniform continuity of the function $f(x)=\operatorname {arctg}x$ ( $-\infty <x<+\infty$ ) on the given domain.

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Problem 799

Investigate the uniform continuity of the function $f(x)=\sqrt{x}$ ( $1\leqslant x<+\infty$ ) on the indicated domain.

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Problem 800

Investigate the uniform continuity of the function $f(x)=x\sin x$ ( $0\leqslant x<+\infty$ ) on the indicated domain.

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Problem 801

Prove that the function $f(x)=\dfrac {\left|\sin x\right|}{x}$ is uniformly continuous on each of the intervals

J_{1}=(-1<x<0) \quad \text{and} \quad J_{2}=(0<x<1)

but is not uniformly continuous on their union

J_{1}+J_{2}=\left\{ 0<\left|x\right|<1\right\}

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Problem 801.1

[NO TASK]

Problem 802

For $\varepsilon >0$ , find $\delta =\delta (\varepsilon )$ (any such value!) for which the function $f(x)$ satisfies the condition of uniform continuity on the indicated interval, where:

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(a)

$f(x)=5x-3$ ( $-\infty <x<+\infty$ );

(b)

$f(x)=x^{2}-2x-1$ ( $-2\leqslant x\leqslant 5$ );

(c)

$f(x)=\dfrac {1}{x}$ ( $0.1\leqslant x\leqslant 1$ );

(d)

$f(x)=\sqrt{x}$ ( $0\leqslant x<+\infty$ );

(e)

$f(x)=2\sin x-\cos x$ ( $-\infty <x<+\infty$ );

(f)

$f(x)=x\sin \dfrac {1}{x}$ ( $x\neq 0$ ) and $f(0)=0$ ( $0\leqslant x\leqslant \pi$ ).

Problem 803

Into how few equal parts must the closed interval $\left[1,10\right]$ be divided so that the oscillation of the function $f(x)=x^{2}$ on each of the resulting segments is less than 0.0001?

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Problem 804

Prove that the sum and the product of a finite number of uniformly continuous functions on an interval $(a,b)$ are still uniformly continuous on that interval.

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Remark.

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Remark: when $(a,b)$ is an infinite interval, the sum $f(x)+g(x)$ of two functions $f(x)$ and $g(x)$ uniformly continuous on $(a,b)$ must also be uniformly continuous, but the product $f(x)g(x)$ need not be uniformly continuous. For example, let $(a,b)=(-\infty ,+\infty )$ ; the function $f(x)=x$ is uniformly continuous on $(-\infty ,+\infty )$ , but the function $\left[f(x)\right]^{2}=x^{2}$ is not uniformly continuous on $(-\infty ,+\infty )$ (see Problem 793(б)).

Problem 805

Prove that if a monotone bounded function $f(x)$ is continuous on a finite or infinite interval $(a,b)$ , then this function is uniformly continuous on $(a,b)$ .

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Problem 806

Prove that a function $f(x)$ defined and continuous on the finite interval $(a,b)$ can be extended by continuity to the closed interval $\left[a,b\right]$ if and only if the function $f(x)$ is uniformly continuous on the interval $(a,b)$ .

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Problem 806.1

[NO TASK]

Problem 807

The function

\omega _{f}(\delta )=\sup \left|f(x_{1})-f(x_{2})\right|

(where $x_{1}$ and $x_{2}$ are any two points in $(a,b)$ subject to the condition $\left|x_{1}-x_{2}\right|\leqslant \delta$ ) is called the modulus of continuity of the function $f(x)$ on the interval $(a,b)$ .

Prove that the function $f(x)$ is uniformly continuous on the interval $(a,b)$ if and only if

\lim _{\delta \to +0}\omega _{f}(\delta )=0 .

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Problem 808

(a)(b)(c)

Let:

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(a)

$f(x)=x^{3}$ ( $0\leqslant x\leqslant 1$ );

(b)

$f(x)=\sqrt{x}$ ( $0\leqslant x<a$ ) and ( $a<x<+\infty$ );

(c)

$f(x)=\sin x+\cos x$ ( $0\leqslant x\leqslant 2\pi$ ).

For the modulus of continuity $\omega_{f}(\delta )$ of the function $f(x)$ (see the preceding problem), give an estimate of the form

\omega _{f}(\delta )\leqslant C\delta ^{\alpha } ,

where $C$ and $\alpha$ are constants.