If $(f_n)$ is a sequence of real functions on the interval $[0, 1]$ converging uniformly to a function $f$ on $[0, 1]$, and if each $f_n$ is continuous except at countably many points, does it follow that there exists a point at which $f$ is continuous?
If $(f_n)$ is a sequence of differentiable functions on $[0, 1]$ converging uniformly to a function $f$ on $[0, 1]$, does it follow that there exists a point at which $f$ is differentiable?
I'm trying to see if I can prove or give counter-examples to the above. I can't see a way to prove them, so I am looking for counter-examples.