Let $f: [0, 1] \to \mathbb R$ be a function on the unit interval. We say $y \in \mathbb R$ is a local maximum value of $f$ if $y = f(x)$ for some strict local maximum $x$ of $f$.
Question: Does there exist a $C^1$ function $f$ on the unit interval such that the set of local maximum values of $f$ is dense in some (nontrivial) open interval $(a, b)$?
Remark: Note that a $C^0$ example is provided by a Brownian motion sample path, since a Brownian motion almost surely achieves a strict maximum in every interval.