differentiability and compactness I have no idea how to show whether this statement is false or true:

If every differentiable function on a subset $X\subseteq\mathbb{R}^n$ is bounded then $X$ is compact.

Thank you

If $X$ is unbounded, take some variant of $\|x\|$.  If $X\subset \mathbb{R}^n$ is bounded but not closed, then there is some $x\in\partial X\cap X'$.  To have any smooth functions, $X$ must have interior, so take a small ball $B\subset X$ with $x\in\partial B$.
Now your job boils down to finding a function on the unit ball which goes to $\infty$ as $x\to (1,0,\cdots,0)$ and goes to $0$ as $x\to$ any other point on the boundary.
