$f'$ is bounded $\Rightarrow$ $f$ is bounded? 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.)
 A: 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.
A: Consider f(x)=x. Over $\mathbb{R}$, it is unbounded but the derivative is constant(and hence bounded). However if you take a finite interval (or an interval contained in a compact set), then the function will be bounded NOT because the derivative is bounded but because the function is continuous.
Edit: I stand corrected. Given continuity and an open interval, a function can be unbounded. Differentiability, same story. Bounded derivative on the other hand will indeed imply boundedness of the function.
