fundamental theorem of calculus for Kurzweil-Henstock integral

Let the symbol denote the Kurzweil-Henstock integral. We can then give the most general version of the fundamental theorem of calculusMathworldPlanetmathPlanetmath.


Let F:[a,b]R and suppose the derivativePlanetmathPlanetmath F(x) exists for all x[a,b]. Then


The reader should note the subtle differencePlanetmathPlanetmath from the standard version. Here we do not assume anything about F except that it exists. For the standard version we usually assume that F is continuousMathworldPlanetmathPlanetmath, and if we use the Lebesgue integralMathworldPlanetmath we must assume that F is Lebesgue integrable. Part of this theoremMathworldPlanetmath is that F is Kurzweil-Henstock integrable, hence no extra assumptionsPlanetmathPlanetmath are necessary.

An example of a function where the standard version has problems is the function

F(x):={x2sin1x2 if x00 if x=0.

F is differentiableMathworldPlanetmathPlanetmath everywhere, but

F(x)={2xsin1x2-2xcos1x2 if x00 if x=0.

Which is not continuous and in fact unboundedPlanetmathPlanetmath on any interval containing zero.

Title fundamental theorem of calculus for Kurzweil-Henstock integral
Canonical name FundamentalTheoremOfCalculusForKurzweilHenstockIntegral
Date of creation 2013-03-22 16:44:27
Last modified on 2013-03-22 16:44:27
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 4
Author jirka (4157)
Entry type Theorem
Classification msc 26A42
Related topic FundamentalTheoremOfCalculusClassicalVersion