Blumberg theorem

In mathematics, the Blumberg theorem states that for any real function ${\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} }$ there is a dense subset D of ${\displaystyle \mathbb {R} }$ such that the restriction of f to D is continuous.

For instance, the restriction of the Dirichlet function (the indicator function of the rational numbers ${\displaystyle \mathbb {Q} }$) to ${\displaystyle \mathbb {Q} }$ is continuous, although the Dirichlet function is nowhere continuous.

Blumberg spaces

More generally, a Blumberg space is a topological space X for which any function ${\displaystyle f:X\rightarrow \mathbb {R} }$ admits a continuous restriction on a dense subset of X. Blumberg theorem therefore asserts that ${\displaystyle \mathbb {R} }$ (equipped with its usual topology) is a Blumberg space.

If X is a metric space, then X is a Blumberg space if and only if it is a Baire space.

References

• Blumberg, Henry (1922). "New properties of all real functions" (PDF). Proceedings of the National Academy of Sciences. 8 (1): 283-288.
• Blumberg, Henry (1922). "New properties of all real functions". Transactions of the American Mathematical Society. 24: 113-128.
• Bradford, J. C.; Goffman, Casper (1960). "Metric spaces in which Blumberg's theorem holds". Proceedings of the American Mathematical Society. 11: 667-670.
• White, H. E. (1974). "Topological spaces in which Blumberg's theorem holds". Proceedings of the American Mathematical Society. 44: 454-462.
• https://www.encyclopediaofmath.org/index.php/Blumberg_theorem