# Proving without the Axiom of Choice (A.C.) that increasing real functions have countable discontinuities

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:

1. Picking a rational at each discontinuity

2. 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

3. 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.

• My idea was like this: if there were uncountable discontinuities in an interval $[n,n+1]$, then $f(n+1)-f(n)\geq \sum_{x\text{ is discontinuity point in }[n,n+1]} (f(x+)-f(x-))=\infty$ and reach a contradiction. Feb 19, 2022 at 0:12
You don't need AC for the proof n°1 (or n°3, I'm not exactly sure what's distinguishing them) because the rationals are well-orderable (this can be done without AC), therefore you can consider a well-ordering $$(\Bbb Q,\preceq)$$ and assign to each jump discontinuity the $$\preceq$$-least rational in $$(\sup_{xc}f(x))$$.
Partition $$\mathbb{R}$$ into disjoint bounded intervals $$(I_m)_{m\geqslant1}.$$ For example, $$[0, 1), [-1, 0), [1, 2), \ldots$$; the details don't matter. Take any strictly decreasing sequence $$(c_n)_{n\geqslant1}$$ with limit $$0$$ (again, the details don't matter), and partition $$\mathbb{R}_{>0}$$ into the disjoint intervals $$J_1 = [c_1, +\infty),$$ $$J_n = [c_n, c_{n-1})$$ for $$n \geqslant 2.$$ Then for all $$m \geqslant 1$$ and all $$n \geqslant 1,$$ the set $$\{x \in \mathbb{R} : x \in I_m, \ f(x+) - f(x-) \in J_n\}$$ is finite. Every point of discontinuity belongs to one of these finite sets, which can be arranged into a sequence, for example by a diagonal traversal.