Let $f : [0,1] \to \left\{ 0, 1 \right\}$ be a function that has at each point a discontinuity of the second kind. Is $f$ measurable if we equip the domain with the Borel or even Lebesgue $\sigma$-algebra and the range with the discrete $\sigma$-algebra?
The indicator of the Cantor set is not a counterexample since it is not discontinuous everywhere. The indicator of the rationals is nowhere continuous and is measurable.
This can be helpful in order to search for counterexamples: If $A \subseteq [0,1]$ is a set such that both $A$ and $[0,1] \setminus A$ are dense in $[0,1]$ then the indicator of $A$ is nowhere continuous. So, the question can be weakened: Is every such set $A$ necessarily measurable?
Wikipedia says that the set of discontinuities is an $F_\sigma$ set, thus a countable union of closed sets and therefore Borel measurable. In our case it is the whole interval $[0,1]$, so this statement doesn't help.
Using the axiom of choice we can construct a Bernstein set, which is a set $A\subseteq[0,1]$ such that no perfect set is a subset of $A$. The construction is such that $[0,1]\setminus A$ is also a Bernstein set, and $A$ meets every perfect set, but contains none. We list all the perfect sets (there are $2^{\aleph_0}$ of them, and we use the smallest possible order type) then we choose from each one two points, one which will enter $A$ and the other which will not. At each step we only chose $<2^{\aleph_0}$ points, so we can continue and choose new points at each step. This ensures that $A$ and its complement are both meeting every perfect subset of $[0,1]$ (and therefore neither contains any).
In particular $A$ is dense co-dense and not measurable. Not Lebesgue and certainly not Borel measurable.