Let \(f_n\colon U\to \mathbb{R}\) be a sequence of functions which converge uniformly to \(f\). If \(f_n\) are all continuously differentiable and \(f_n'\) converge locally uniformly to some function \(g\) then \(f\) is also continuously differentiable with \(f'=g\).
Remark
- The same is true for functions with codomain \(\mathbb{R}^m\) or \(\mathbb{C}^m\) and domain \(\mathbb{R}^n\) or \(\mathbb{C}^n\).