# 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 FundamentalTheoremOfCalculusForKurzweilHenstockIntegral 2013-03-22 16:44:27 2013-03-22 16:44:27 jirka (4157) jirka (4157) 4 jirka (4157) Theorem msc 26A42 FundamentalTheoremOfCalculusClassicalVersion