Let $f: \mathbb R \to \mathbb R$ be a locally integrable measurable function.
We say $f$ satisfies the intermediate value property if given any $a, b\in \mathbb R$ with $a < b$, whenever $u \in \mathbb R$ is such that $\min(f(a), f(b)) \leq u \leq \max(f(a), f(b))$, there exists some $x \in [a, b]$ such that $f(x) = u$.
Question: Suppose every point of $\mathbb R$ is a Lebesgue point of $f$. Does it follow that $f$ satisfies the intermediate value property?
Note: We use the “strong” definition of Lebesgue point, given here.
Best Answer
If $F(x)=\int_a^x f(x)dx$ (for some fixed $a$), then $x$ being a Lebesgue point of $f$ yields $F'(x)=f(x)$; and the derivatives enjoy the intermediate value property by Darboux theorem.