Is the set of discontinuity of $f$ countable? Suppose $f:[0,1]\rightarrow\mathbb{R}$ is a bounded function satisfying: for each $c\in [0,1]$ there exist the limits $\lim_{x\rightarrow c^+}f(x)$ and $\lim_{x\rightarrow c^-}f(x)$. Is true that the set of discontinuity of $f$ is countable?

Yes as $f$ must be a regulated function and hence only has countable many discontinuities.
A regulated function is a function which has a right and a left hand limit. This is equivalent to that there is a sequence of step functions converging uniformly towards the function. As a step function only has finite many discontinuies, the regulated function can only have countable many discontinuies.
