Exercise 12, § 6

Let $f$ be a continuous non-constant finite real-valued function on a compact interval $[a, b]$ of $\mathbf{R}$, such that $f(a) = f(b) = 0$; let $Z$ be the closed subset of $[a, b]$ consisting of points $x$ for which $f(x) = 0$, and let $c$ be the largest of the lengths of the intervals contiguous to $Z$ in $[a, b]$. Show that, for each $t$ such that $0 < t < c$, there is a point $x \in [a, b]$ such that $x + t \in [a, b]$ and $f(x + t) = f(x)$.