MathDB

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

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