Is the preimage of a $[a, b)$ interval under a continuous function $ f: \mathbb{R} \rightarrow \mathbb{R} $ always with measure zero boundary?

Question

I'm trying to prove a result in probability theory and I would really like to find a way to approximate a continuous bounded function with a succession of linear combinations of indicator functions of Borel sets with measure zero boundary. Maybe you can see my reasoning. But I got stuck on the question above.

Answer 1

No. Take a "fat Cantor set" $C \subset \mathbb R$ (see for example https://en.wikipedia.org/wiki/Smith%E2%80%93Volterra%E2%80%93Cantor_set). This is a nowhere dense compact set with positive measure. Define $$f : \mathbb R \to \mathbb R, f(x) = -d(x,C)$$ where $d(x,C) = \inf_{c\in C} d(x,c)$. This is a continuous function such that $f(\mathbb R) = (-\infty,0]$. We have $f^{-1}([0,1)) = f^{-1}(0) = C$. Since $C$ is nowhere dense, it has empty interior and thus $C = \text{bd} (C)$.