习题 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 $[a, b]$. 证明，对于每个$t$使得$0 < t < c$，存在点$x \in [a, b]$使得$x + t \in [a, b]$和$f(x + t) = f(x)$。