Does differentiability on an open set imply differentiability on a compact set?

Question

If I have a function $f(x) \in C^\infty$ for $x \in (-\epsilon,+\epsilon)$. Can I find a compact set in $(-\epsilon,+\epsilon)$ where $f(x)$ is also $\in C^\infty$? I need this information to imply that $f(x)$ is bounded in that compact set, so it takes it´s maximum/minimum.

Answer 1

So if a function $f(x)$ is continuous differentiable on an open set $(-\epsilon, \epsilon)$ then there exists an compact set such as $[-\epsilon/2,\epsilon/2]$ in which $f(x)$ is continuously bounded and takes its maximum.