Every real function has a dense set on which its restriction is continuous

Question

The title says it all: if $f\colon \mathbb{R} \to \mathbb{R}$ is any real function, there exists a dense subset $D$ of $\mathbb{R}$ such that $f|_D$ is continuous.

Or so I'm told, but this leaves me stumped. Apart from the rather trivial fact that one can find a dense $D$ such that the graph of $f|_D$ has no isolated points (by a variant of Cantor-Bendixson), I don't know how to start. Is this a well-known fact?

score 49 · Accepted Answer · answered Aug 13 '11 at 00:13

49

It is a theorem due to Blumberg (New Properties of All Real Functions - Trans. AMS (1922)) and a topological space $X$ such that every real valued function admits a dense set on which it is continuous is sometimes called a Blumberg space.

Moreover, in Bredford & Goffman, Metric Spaces in which Blumberg's Theorem Holds, Proc. AMS (1960) you can find the proof that a metric space is Blumberg iff it's a Baire space.

answered Aug 13 '11 at 00:13

Samuele

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Blumberg's paper is online: http://www.ams.org/journals/tran/1922-024-02/S0002-9947-1922-1501216-9/S0002-9947-1922-1501216-9.pdf – Autumn Kent Aug 13 '11 at 00:18
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Thanks for the reference! For completeness, there's a self-contained and rather nice proof (or at least nicer than Blumberg's original one) of the statement, plus an extensive discussion, in the more general setting, in chapter 8 of Goffman, Nishiura, and Waterman's book Homeomorphisms in Analysis, available online at http://www.ams.org/publications/online-books/surv54-index – Gro-Tsen Aug 13 '11 at 23:09

Every real function has a dense set on which its restriction is continuous

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