numerical analysis
For problem 1, the proof is obvious by definition. If $\displaystyle f\in C^n(\Omega)$ then $\displaystyle f^\prime, f^{\prime\prime}, \ldots, f^{(n)}$ are all continuous. Let $\displaystyle g = f'$. Now, what are $\displaystyle f^{\prime\prime}, \ldots, f^{(n)}$ in terms of $\displaystyle g$? What does that tell you about $\displaystyle g$'s differentiability class?
I have no ideas for problems 2-4.