I encountered the following problem in Berkeley problems in Mathematics:
(Sp84): Prove or supply a counterexample: If the function $f$ from $\mathbb{R}$ has both a left limit and a right limit at each point of $\mathbb{R}$, then the set of discontinuities of $f$ is, at most, countable.
I found the book claimed this is right. But I have the following counterexample:
Let $f=0$ at $\mathbb{R}$ except at the cantor set. And let $f=1$ at the cantor set. Then $f$ has both a left limit and a right limit at every point in $\mathbb{R}$. But the set of discontinuities is the cantor set, whose cardinality is equal to $c$.
This should make sense since the Cantor set is nowhere dense, and the left/right limit at every point should be 0. I just do not know why this counterexample does not make sense - or maybe the book means all non-removable discontinuities?