SCHWEITZER 1981 Problem 8

Let $W$ be a dense, open subset of the real line $\mathbb{R}$ . Show that the following two statements are equivalent:

(1) Every function $f : \mathbb{R} \rightarrow \mathbb{R}$ continuous at all points of $\mathbb{R} \setminus W$ and nondecreasing on every open interval contained in $W$ is nondecreasing on the whole $\mathbb{R}$ .

(2) $\mathbb{R} \setminus W$ is countable.

E. Gesztelyi