I wanted to ask the following question: If f is Riemann integrable in [a, b], is there necessarily a point in [a, b] where f is continuous? I know there is a theorem that states that f is integrable if and only if its set of discontinuities is measure 0. Does that imply that in a closed interval, f must have a point of continuity? Or is this not necessarily true? Thanks in advance
2026-03-28 08:38:08.1774687088
Does an integrable function have to have a point of continuity?
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An interval has Lebesgue measure equal to its length, so an entire genuine interval (not $(a,a) $ or $[a,a] $) cannot be contained in the set of discontinuities of a Riemann-integrable function.