This question already has an answer here:
$f:\mathbb R\rightarrow \mathbb R$ 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 $?$