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Also fun is the Dirichlet function[1], which is discontinuous at every point yet is integrable everywhere. There's also a similar function that is discontinuous
by rcthompson 13y ago
Also fun is the Dirichlet function[1], which is discontinuous at every point yet is integrable everywhere. There's also a similar function that is discontinuous at all rational numbers but continuous at all irrational numbers[2].
[1] http://en.wikipedia.org/wiki/Dirichlet_function http://en.wikipedia.org/wiki/Dirichlet_function
[2] http://en.wikipedia.org/wiki/Thomae%27s_function http://en.wikipedia.org/wiki/Thomae%27s_function
- alowe 13y agoIn a similar vein, the cantor function [1] (also known as the devil's staircase) is continuous and has derivative 0 almost everywhere, but still manages to increase in value. [1] http://en.wikipedia.org/wiki/Cantor_function#Properties http://en.wikipedia.org/wiki/Cantor_function#Properties
- suppaman 13y agoI believe that is only true when you consider certain types of integration. E.g. True for Lebesgue, false for Riemann
- ska 13y agoIn analysis, people generally mean Lebesgue integrable unless otherwise specified.