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 $F\mathrm{:}\mathrm{[}a\mathrm{,}b\mathrm{]}\mathrm{\to }\mathrm{R}$ and suppose the derivative$F^{\mathrm{\prime }}\mathit{}\mathrm{(}x\mathrm{)}$ exists for all $x\mathrm{\in }\mathrm{[}a\mathrm{,}b\mathrm{]}$. 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

$F$ is differentiable everywhere, but

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