Set of discontinuities of a function that has both limits at each point of $R$

Question

$f:\mathbb R\rightarrow \mathbb R$ has both a left limit and a right limit at each point of $\mathbb R.$ Then the number of discontinuities of $f$ is what $?$

Now the Greatest Integer Function is one such function and has countable infinite discontinuities.

Any continuous function also has both limits at each point and they are same and number of discontinuity is $0.$

So , is there a function with both limits at each point that has number of discontinuities uncountable $?$ Is that possible $?$

Answer 1

The number of discontinuities of such an $f $ is at most countable. Indeed, since the lateral limits always exist, every discontinuity is a jump one; and the number of jump discontinuities is at most countable.