I know three different but similar proofs of the statement:
If $f:\mathbb{R}\to\mathbb{R}$ is an increasing function, then there are at most countably many discontinuities.
But each of the proofs relies on A.C. Therefore I am wondering if there is a way to prove this without using A.C.
The three proofs I know are as follows:
Picking a rational at each discontinuity
Proving that there are at most countably many discontinuities in each interval $[n,n+1]$, and then showing that this countable union of countable sets is countable
Using that there are at most countably many disjoint open intervals in $\mathbb{R}$, and that $(f(x-),f(x+)),x\in\{f\text{ is discontinuous at }x\}$ is a collection of disjoint intervals.
Please advise me about this!