When discussing this question, we made interesting observations on increasing functions with discontinuities on rationals. Let $f: [0,1] \to [0,1]$ be an increasing function with discontinuities on rationals (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 densely ordered subset of $[0,1]$. When is the image of $f$ is a Borel uncountable set and when is a null set?
I guess that the first holds under mild conditions and the second holds if the image is doubly bounded