= Solution
{c}
An explicit <piecewise linear function> is
$$
\boxed{f(t)=\begin{cases}
-3t,&0\leq t\leq1/3,\\
6t-3,&1/3\leq t\leq2/3,\\
3-3t,&2/3\leq t\leq1.
\end{cases}}
$$
The adjacent formulas agree at $1/3$ and $2/3$, so the <function> is <continuous>, and $f(0)=f(1)=0$. Any pair of points at distance $2/3$ can be written $t,t+2/3$ with $0\leq t\leq1/3$. For every such $t$,
$$
f(t+2/3)-f(t)=\bigl(1-3t\bigr)-(-3t)=1.
$$
Thus their values are never equal. The <horizontal chord of reciprocal-integer length> conclusion does not extend to this distance.
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