Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $I.$

share|improve this question

4 Answers 4

up vote 6 down vote accepted

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.

share|improve this answer

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.

share|improve this answer

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.

share|improve this answer
    
Hmmm, but I have task to prove it:( Maybe $I$ matter here? For, example it is closed. –  RiaD Mar 10 '12 at 22:03
    
I think it would matter, give me some time to work out the details. –  user21436 Mar 10 '12 at 22:08
    
We need a function for which $f'$ exists everywhere. –  Hans Engler Mar 11 '12 at 5:00
    
$f'$ does exist everywhere on $(0,1]$ @HansEngler –  user21436 Mar 11 '12 at 5:04
1  
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. –  Hans Engler Mar 11 '12 at 5:18

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 $\delta$.

share|improve this answer
    
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 $0$. –  chharvey Feb 1 '13 at 15:16

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.