Preimage of continuous function of intervall

Question

Let $a<b\in \mathbb{R}$ and $f:]a,b]\to\mathbb{R}$ a continuous function. Is the preimage of every closed intervall $I$ representable as a countable union of intervalls? If yes, is there a special type of intervalls (e.g. half-open) of which the union can be represented?

Answer 1

Let $C$ be the Cantor set (suitably stretched & translated to be a subset of $(a,b]$) and $f(x) = \min_{c \in C} |x-c|$. Then $f$ is continuous and $f^{-1}([-1,0]) = C$.

Since $C$ is uncountable and contains no interior points, it cannot be represented as a countable union of any sort of interval.