Let be $(X, d)$ a complete metric space, $f_n : X \rightarrow \mathbb{R}$ a sequence of continuous functions and a function $f: X \rightarrow \mathbb{R}$ such that: $$\lim_{n \rightarrow +\infty} f_n(x)=f(x), \ \ \ \ \ \ \ \ x \in X$$ Prove that:
- exist and open dense set $W$ of $X$ such that $f$ is locally bounded in $W$
- the set $\{ x \in X :\ f \ continuous\ at\ x\}$ is a $G_\delta$ dense set in $X$. Can the Dirichlet function at x be the limit of a sequence of continuous functions $f_n \ $ (pointwise convergence)? Hence can the derivative of a differentiable function be not continuous in a interval $I \subset \mathbb{R}$ where its interior is not empty ( $\mathring I \neq \emptyset$)?
The function $f$ such defined is a Baire-1 function, and I'm required to prove these statements. The exercise suggests to use the fact that: $$X=\bigcup_{k\geq1}\bigcap_{n\geq1}\{|f_n|\leq k\}$$ and repeat the statement on closed-balls inside $X$, but I don't know how to do it.