Problem 1431 (difficulty: 7/10)

Prove that if \(\displaystyle f:\R \to \R\), then the set of points of continuity is Borel, and give as small as possible of Borel-class (e.g.. \(\displaystyle G_{\delta\sigma\delta\sigma\delta\sigma\delta\sigma}\)), to which it still belongs.

Solution:

For every positive integer \(\displaystyle n\) let

\(\displaystyle \mathcal{I}_n = \Big\{ I\subset\R:\text{~$I$ is an open interval and~} \sup\limits_If-\inf\limits_If<\tfrac1n \Big\} \quad\text{and let}\quad A_n = \cup\mathcal{I}_n = \bigcup_{I\in\mathcal{I}_n}I. \)

By Cauchy's criterion, any \(\displaystyle a\in\R\) is a point of continuity of \(\displaystyle f\) if and only if

\(\displaystyle \forall n\in\N ~~ \exists I\in \mathcal{I}_n ~~ a\in I, \)

or equivalently

\(\displaystyle \forall n\in\N ~~ a \in A_n. \)

Therefore, the set of points of continuity is \(\displaystyle \bigcap\limits_{n\in\N} A_n\), that is in \(\displaystyle G_\delta\).

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