Patrick M. Fitzpatrick, Brian R. Hunt
Abstract
We describe a proof of the fundamental theorem of calculus for the Lebesgue integral, based on the following result: A real-valued function f on a compact interval is absolutely continuous if and only if its family of divided difference functions, {x ↦ [f(x + h) − f(x)]/h}0
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We describe a proof of the fundamental theorem of calculus for the Lebesgue integral, based on the following result: A real-valued function f on a compact interval is absolutely continuous if and only if its family of divided difference functions, {x ↦ [f(x + h) − f(x)]/h}0
Key concepts: Absolute continuity, Mathematics, Uniform limit theorem, Lebesgue integration, Pointwise convergence, Locally integrable function, Uniform convergence, Pointwise