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I've $I = [0 ,+\infty)\,$ and $f: I \rightarrow \Bbb R.$

a. I've proved that if $f'$ is bounded on $I$ then $f$ is uniformly continuous on $I$.
b. I've proved that if $\lim f' = \infty$ (with $x \rightarrow +\infty$) then $f$ isn't uniformly continuous on $I$.

c. Now I should prove that if $f'$ is unbounded on $I,$ then isn't uniformly continuous on $I$.
Using b I've proved that if c is wrong, there is a segement $T = [0, t]$ where $f'$ is unbounded.

Added: the also known that $f'$ exists on every point on $I.$

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4 Answers 4

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As @Kannappan suggested, here is my previous comment, expanded into an answer.

Assertion c) is indeed not correct, not even for bounded intervals. That is, every function that is differentiable on a closed and bounded interval is continuous and hence automatically uniformly continuous there, even if its derivative is unbounded (which is of course possible). An example for the latter case (differentiable everywhere, unbounded derivative on a bounded and closed interval) is $$ f(x) = \begin{cases} x^{3/2} \sin \frac{1}{x} (0 < x \le 1) \\ 0 (x = 0) \end{cases} $$ For the case $I = [0,\infty)$, a counterexample is given by $g(x) = f(\frac{x}{1+x})$ where $f$ is defined as above.

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This is a wrong claim as the answer by Professor Aguirre suggests. I give an example in the case of a neither closed nor open set. It is known to be false even in general as the other answer suggests.


You seem to claim that:

If a function $f$ has unbounded derivative, this does not mean that $f$ is not uniformly continuous.

This is a wrong claim. Here is a counter example.

Consider $f:(0,1] \to \mathbb R$ defined by $f(x)=\sqrt x$. I leave it to you to prove that $f'$, the derivative of $f$ is unbounded on $(0,1]$ but $f$ is uniformly continuous on $(0,1]$.

Conclusion:

What is true is, there are functions with unbounded derivatives nevertheless uniformly continuous.

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  • $\begingroup$ Hmmm, but I have task to prove it:( Maybe $I$ matter here? For, example it is closed. $\endgroup$
    – RiaD
    Commented Mar 10, 2012 at 22:03
  • $\begingroup$ I think it would matter, give me some time to work out the details. $\endgroup$
    – user21436
    Commented Mar 10, 2012 at 22:08
  • $\begingroup$ We need a function for which $f'$ exists everywhere. $\endgroup$ Commented Mar 11, 2012 at 5:00
  • $\begingroup$ $f'$ does exist everywhere on $(0,1]$ @HansEngler $\endgroup$
    – user21436
    Commented Mar 11, 2012 at 5:04
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    $\begingroup$ Let $f(x) = x^{3/2} \sin \frac{1}{x}$ for $x \in (0,1]$ and $f(0) = 0$. One can calculate that $f'$ exists everywhere on $[0,1]$, in particular $f'(0) = 0$, and $f'(\frac{1}{n \pi})$ is unbounded as $n \to \infty$. Yet $f$ is of course uniformly continuous on $[0,1]$ since the interval is compact. The OP's assertion c) therefore is incorrect. @RiaD - go and give them a counterexample and you will henceforth be afforded great respect. $\endgroup$ Commented Mar 11, 2012 at 5:18
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Claim c. is not true in general. I will provide an example in $[\,0,+\infty)$, as asked by the OP. Let $g\colon[\,0,+\infty)\to\mathbb{R}$ be defined as follows. For each $n\in\mathbb{N}$, $g$ is piecewise linear on the interval $$\Bigl[\,2^n-\dfrac{1}{n\,2^n},2^n+\dfrac{1}{n\,2^n}\,\Bigr]$$ and $$ g\Bigl(2^n-\frac{1}{n2^n}\Bigr)=g\Bigl(2\,^n-\dfrac{1}{n2^n}\Bigr)=0,\quad g(2^n)=n. $$ Outside those intervals, $g$ is equal to zero. Let $f(x)=\int_0^xg(t)\,dt$. Then $f$ is increasing and $\lim_{x\to+\infty}f(x)=\sum_{n=1}^\infty2^{-n}=1$. This implies that $f$ is uniformly continuous on $[\,0,+\infty)$, but $f'=g$ is unbounded.

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  • $\begingroup$ just a question, why does the $\lim_{x\to+\infty}f(x)=1$ imply $f$ is uniformly continuous on $[0,+\infty)$? $\endgroup$
    – C Squared
    Commented Nov 7, 2020 at 10:54
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    $\begingroup$ also, kind of nit picky, but do you mean $g(2^n\textbf{+}\frac{1}{n2^n})=0$ in the very middle of the answer? $\endgroup$
    – C Squared
    Commented Nov 7, 2020 at 11:02
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One thing at work here is that, although the derivative might be large, if it doesn't have a sufficiently large interval on which to work, then it doesn't always have a chance to create a large increase in $y$-values.

For example, consider $x^{1/3}$ near the origin and the interval $[0,\delta]$. No matter how close a positive $x$ is to $0$, the maximum change possible is completely dependent on the value $\delta$.

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  • $\begingroup$ Interesting example. What makes $x^{1/3}$ different from, say, $\ln(x)$? Why is the former uniformly continuous whereas the latter is not? They both have unbounded derivatives near $0$. $\endgroup$
    – chharvey
    Commented Feb 1, 2013 at 15:16
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    $\begingroup$ @chharvey $\ln(x)$ is unbounded itself, unlike $x^{1/3}$. $\endgroup$
    – Ruslan
    Commented Feb 5, 2015 at 15:01

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