If a sequence of functions $f_n$ converges uniformly to $f$, does this imply $f_n$ is continuous? How to prove?

Question

I know I can find an $f_n$ that converges to f uniformly, but can we use this to argue $f_n$ is continuous? Is there a theorem I can state?

Answer 1

No. Even if $f$ is continuous and if $(f_n)_{n\in\mathbb N}$ converges uniformly to $f$, the functions $f$ do not to be continuous. Take, for instance$$\begin{array}{rccc}f_n\colon&&\mathbb R\longrightarrow&\mathbb R\\&x&\mapsto&\begin{cases}0&\text{ if }x\neq0\\\frac1n&\text{ if }x=0.\end{cases}\end{array}$$Then $(f_n)_{n\in\mathbb N}$ converges uniformly to the null function, but no $f_n$ is continuous.