Let $f:[0,1]\to\mathbb R$ be an arbitrary function. It is well known that if $f$ is of Baire Class 1, then the points at which $f$ is continuous form a dense subset of $[0,1]$. My question now is, if the converse is true:
Let $f$ be continuous at each point of a dense subset of $[0,1]$. Is then $f$ necessarily of Baire Class 1?
I guess not, and I would like to see some references showing a counter example or, even better, some book/paper answering this and the following question:
If the answer to the above question is negative, what additional property on $f$ is needed to guarantee that $f$ is of Baire Class 1?
I suppose that the set of continuity in a certain sense has to be bigger than just dense.
Any help is highly appreciated. Thank you in advance!