Blumberg theorem
In mathematics, the Blumberg theorem states that for any real function f : ℝ → ℝ there is a dense subset D of 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 ℚ) to 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 f : X → ℝ admits a continuous restriction on a dense subset of X. Blumberg theorem therefore asserts that 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.
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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
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