Let $S$ be a subset of $C^1([0, 1], \mathbb{R})$. It is a well-known fact that given a function $f\in C^1([0, 1], \mathbb{R})$ and a sequence $\{f_n\}\subset C^1([0,1], \mathbb{R})$ such that $f_n\to f$ uniformly, it does not necessarily hold that $f_n'\to f$ uniformly.

What condition on $S$ (or what condition on $f$) should we impose to guarantee the existence of $f_n$ such that both $f_n\to f$ and $f_n'\to f$ in the uniform sense? Are there any non-trivial results to this question?