fundamental theorem of calculus for Kurzweil-Henstock integral

Let the $\int$ symbol denote the Kurzweil-Henstock integral. We can then give the most general version of the fundamental theorem of calculus.

Theorem.

Let $F\colon[a,b]\to{\mathbb{R}}$ and suppose the derivative $F^{\prime}(x)$ exists for all $x\in[a,b]$ . Then

$\int_{a}^{b}F^{\prime}(x)dx=F(b)-F(a).$

The reader should note the subtle difference from the standard version. Here we do not assume anything about $F^{\prime}$ except that it exists. For the standard version we usually assume that $F^{\prime}$ is continuous, and if we use the Lebesgue integral we must assume that $F^{\prime}$ is Lebesgue integrable. Part of this theorem is that $F^{\prime}$ is Kurzweil-Henstock integrable, hence no extra assumptions are necessary.

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

$F(x):=\begin{cases}x^{2}\sin\frac{1}{x^{2}}&\text{ if $x\not=0$}\\ 0&\text{ if $x=0$}.\end{cases}$

$F$ is differentiable everywhere, but

$F^{\prime}(x)=\begin{cases}2x\sin\frac{1}{x^{2}}-\frac{2}{x}\cos\frac{1}{x^{2}% }&\text{ if $x\not=0$}\\ 0&\text{ if $x=0$}.\end{cases}$

Which is not continuous and in fact unbounded 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