Suppose $E:=(E,\|\!\cdot\!\|)$ is a Banach space and $I:=[\alpha,\beta]$ is a compact perfect interval. A function $f:I\to E$ is jump continuous if the one-sided limits $f(\alpha+),f(\beta-)$ and $f(x\pm)$ exist for all $x\in I^\circ$.
The following is Exercise VI.1.5 from Analysis II by Amann & Escher:
Prove that every jump continuous function has at most a countable number of discontinuities.
I know that monotone functions ($\mathbb{R\to R}$) have the same property, whose proof begins by choosing a rational number between $f(x-)$ and $f(x+)$. I tried to imitate, but I don't know what to do when $E$ is not totally ordered as $\mathbb{R}$ is.
The proof of this exercise for the case $E=\mathbb{R}$ can be found here, but I don't know how the proof can be modified for the general case. However, I think the proof here works for the general case, but it does not seem "natural" to me. I'm not familiar with the fact that a discrete subset of $\mathbb{R}$ is countable (which does not appear in Amann & Escher). Also, the exercise comes with no hint, so I think it should be easy in some sense. Maybe I'm missing something. Are there other proofs?