Pether Luthy gave an example of a sequence of continuous real valued functions whose supremum was discontinuous on a set of positive measure. But does it exist a sequence of continuous real valued functions $f_n:\mathbb{R}\to\mathbb{R}$ such that $f(x) = \sup_{n \in \mathbb{N}} f_n(x)$ is a discontinuous function at every point of a subinterval of $\mathbb{R}$ ?

If such a sequence does not exist, how is it possible to prove it?

everypoint in the interval. $\endgroup$