RO-NMO 2004 Problem 113

Let $f : (a,b) \to \mathbb R$ be a function with the property that for all $x \in (a,b)$ there is a non-degenerated interval $[ a_x,b_x ]$ with $a < a_x \leq x \leq b_x < b$ such that $f$ is constant on $\left[ a_x,b_x \right]$ .

(a) Prove that $\textrm{Im} f$ is finite or numerable.

(b) Find all continuous functions which have the property mentioned in the hypothesis.