Exercise 16, § 8

a) Let $A$ be a countable closed subset of $\mathbf{R}$ and let $f$ be a continuous real-valued function defined on $\mathbf{R}$. Show that if $f$ is constant in each of the intervals contiguous to $A$, then $f$ is constant (use Bolzano's theorem).

b) If $f$ is the continuous mapping of Cantor's triadic set $K$ onto $[0, 1]$ defined in Exercise 9, show that $f$ can be extended to a continuous function on $\mathbf{R}$ which is constant on each of the intervals contiguous to $K$.