I am trying to understand why the set of discontinuities of an increasing function $f: \mathbb R \to \mathbb R$ must be finite or countable. I showed that such function can only have jump discontinuities. It is not clear to me why the discontinuities can't be uncountable and it is also not clear to me for given $f$ how to find a bijection between discontinuities of $f$ and a subset of $\mathbb Q$. Please can somebody explain me why it should be true?
At each jump, draw an open interval on the $y$-axis which fits in between the bottom of the jump and the top of the jump.
After you are done, you have a string of disjoint intervals on the $y$-axis (because the function is increasing.)
Now choose a rational number in each of these intervals. This gives you a bijection between the intervals and a subset of $\Bbb Q$.
The main idea is that there is a limit to how big a disjoint collection of open sets in $\Bbb R$ can be. The generalized idea is that of the Souslin number of a topological space. In a nutshell, it's "the largest cardinality of a set of disjoint open sets."
You can see that the same argument works with $\Bbb R$ replaced with any separable linearly ordered topological space.