Horizontal chords of the graph of a continuous function

Abstract:

The continuous function $f$, defined on $[a,b]$, changing sign exactly $n$ times on $(a,b)$, and $f(a) = f(b) = 0$, has horizontal chords of every length $h < H$ with endpoints in $(a,b)$. We determine the largest such $H$ as a function of $n$.