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The Dirichlet function is not Riemann-integrable on any segment of despite being bounded because the set of its discontinuity points is not negligible (for the Lebesgue measure). The Dirichlet function provides a counterexample showing that the monotone convergence theorem is not true in the context of the Riemann integral.
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It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. [ 1 ] [ 2 ] It is well known for its results on prime numbers (involving the Prime Number Theorem and Riemann zeta function ) and additive number theory (such ...
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The Dirichlet function, which is the indicator function for rationals, is a bounded function that is not Riemann integrable. The Cantor function is a monotonic continuous surjective function that maps [ 0 , 1 ] {\displaystyle [0,1]} onto [ 0 , 1 ] {\displaystyle [0,1]} , but has zero derivative almost everywhere .
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