proof of continuous functions are Riemann integrable

Recall the definition of Riemann integral. To prove that f is integrable we have to prove that limδ0+S*(δ)-S*(δ)=0. Since S*(δ) is decreasing and S*(δ) is increasing it is enough to show that given ϵ>0 there exists δ>0 such that S*(δ)-S*(δ)<ϵ.

So let ϵ>0 be fixed.

By Heine-Cantor Theorem f is uniformly continuousPlanetmathPlanetmath i.e.


Let now P be any partitionPlanetmathPlanetmath of [a,b] in C(δ) i.e. a partition {x0=a,x1,,xN=b} such that xi+1-xi<δ. In any small interval [xi,xi+1] the function f (being continuousMathworldPlanetmathPlanetmath) has a maximum Mi and minimum mi. Since f is uniformly continuous and xi+1-xi<δ we have Mi-mi<ϵ/(b-a). So the differencePlanetmathPlanetmath between upper and lower Riemann sums is


This being true for every partition P in C(δ) we conclude that S*(δ)-S*(δ)<ϵ.

Title proof of continuous functions are Riemann integrable
Canonical name ProofOfContinuousFunctionsAreRiemannIntegrable
Date of creation 2013-03-22 13:45:34
Last modified on 2013-03-22 13:45:34
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 7
Author paolini (1187)
Entry type Proof
Classification msc 26A42