Why if $f'$ is unbounded, then $f$ isn't uniformly continuous?

Question

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 $I.$

Answer 1

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.

Answer 2

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.

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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.

Answer 3

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.

Answer 4

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 $\delta$.