Say I have a function $f:\mathbb{R} \to \mathbb{R}$ which is continously differentiable and has a bounded derivative. Then I know I can approximate $f$ with smooth functions $\phi_n$ by mollifications such that $\phi_n \to f$ uniformly. My question is then, does the convergence$$\phi_n' \to f'$$hold uniformly on compact sets? I know it holds pointwise, but I'm not sure where to begin with a proof/counter example of the uniform convergence.
↧