When discussing this question, we made interesting observations on increasing (i.e., nondecreasing) functions with discontinuities on rationals. Let $f: [0,1] \to [0,1]$ be an increasing right-continuous function with discontinuities on rationals whose jump values sum to $1$ (see here for examples). It can be shown that this function is strictly (monotonically) increasing, which next implies that its image is a null uncountable set (see this answer). The image of $f$ is Lebesgue measurable, and it seems it is also Borel. This motivates me to ask this question:
Let $f: [0,1] \to \mathbb R$ be an increasing function with discontinuities on a countable set that is dense in $[0,1]$. When is the image of $f$ a Borel uncountable set and when is a null set?
I guess that the first holds under mild conditions. I am also wondering how the result changes if $f(0)$ and $f(1)$ are not necessarily finite numbers and the function can tend to infinity at the boundaries, which allows us to extend the result to the case that $\text{dom}(f)$ is not necessarily a closed and bounded interval (I think this case can be handled by applying a strictly increasing continuous transform to $f$ that maps its image to a bounded interval).