TITLE: $f'$ is bounded $\Rightarrow$ $f$ is bounded?
QUESTION [5 upvotes]: Let $I$ be a bounded connected subset in $\mathbb{R}$ and $f: I\rightarrow \mathbb{R}^k$ be a differentiable function.
Does boundedness of $f'$ imply boundedness of $f$?
(I edited this post after I realized that I didn't actually write what i wanted, after i saw Gautam's post.)

REPLY [5 votes]: If you mean by interval a bounded interval, then the answer is yes because of the mean value theorem. If not then the identity function provides a counterexample.
Added explanation of the bounded interval case. We may assume (by adding a constant which does not affect boundedness of $f$ and even less that of $f'$) that $f(x_0)=0$ for some $x_0\in I$. The interval has some finite length $l$, and the mean value theorem implies that whenever $|f(x)|>lC$ for some $C$ then there exists $x'\in I$ such that $|f'(x')|=\frac{|f(x)|}{|x-x_0|}>C$, so that if $f$ is unbounded, $f'$ must also be unbounded.