Let \(f_n\colon \mathbb{R}\to \mathbb{R}\) be a sequence of functions which converge locally uniformly to \(f\). If \(f_n\) are all continuous , so is \(f\).

Remark
  • The same is true for functions with codomain \(\mathbb{R}^m\) or \(\mathbb{C}^m\) and domain \(\mathbb{R}^n\) or \(\mathbb{C}^n\).