Generalising 'if a sequence of continuous functions tend pointwise on $[0,1]$ to f then f has a point of continuity'

Question

Using the Baire Category Theorem I have proven that if a sequence of continuous real-valued functions defined on [0,1] tend pointwise to a function f, then f has a point of continuity.

Is it valid to generalise this by arguing that, on any interval $[a,b] \subset [0,1]$ the same argument applies, and we have a point of continuity, and so f's points of continuity must be dense on $[0,1]$

Answer 1

Yes, it is true that the set of points of continuity of the point-wise limit of sequence of continuous functions is dense.