# Pointwise convergence of sequence of continuous functions

The very first example of pointwise convergence of continuous functions one meets is $f_n(x)=x^n$ with $x\in [0,1]$.

Here $f_n$'s are continuous on $[0,1]$, but their pointwise limit is not continuous at $1$.

Q. Is there an example of sequence $f_n$ of continuous functions from $[0,1]$ to $\mathbb{R}$ such that their pointwise limit is nowhere continuous?

• I think the Dirichlet function would help: en.wikipedia.org/wiki/Nowhere_continuous_function
– ntt
Apr 10 '17 at 5:23
• No, the limit function will be continuous on a dense $G_\delta$ subset of $[0,1]$ (which must be uncountable btw). This is a consequence of Baire's theorem.
– zhw.
Apr 10 '17 at 5:29

$$f_n(x)=\begin{cases} 1+\dfrac{x}{n}, \: x\in [0,1]\cap\mathbb{Q}\\0,\: \text{otherwise}\end{cases}$$
• Is $f_n$ continuous on whole $[0,1]$? Apr 10 '17 at 5:30