Here is one example. Let $(q_n)_n$ be a list containing all rationals in $[0,1]$, let $f_n:[0,1]\to\mathbb{R}$ be given by $f_n(x)=1$ if $|x-q_n|<\frac{1}{n!}$ and $f_n(x)=0$ if not. Let $f=\sum_nf_n:[0,1]\to\mathbb{R}$.
If some function $g$ is in $[f]$ then it has to be unbounded in any open interval, so it cannot be continuous at any point.