PlanetMath: fundamental theorem of calculus for Kurzweil-Henstock integral

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fundamental theorem of calculus for Kurzweil-Henstock integral

(Theorem)

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

Theorem 1 Let $F \colon [a,b] \to {\mathbb{R}}$ and suppose the derivative exists for all $x \in [a,b]$ . Then

$\displaystyle \int_a^b F'(x) dx = F(b)-F(a) .$

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

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

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

is differentiable everywhere, but

$\displaystyle F'(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.

"fundamental theorem of calculus for Kurzweil-Henstock integral" is owned by jirka.

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Cross-references: interval, unbounded, differentiable, function, necessary, Lebesgue integrable, Lebesgue integral, continuous, difference, derivative, fundamental theorem of calculus, Kurzweil-Henstock integral
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This is version 1 of fundamental theorem of calculus for Kurzweil-Henstock integral, born on 2007-02-23.
Object id is 8964, canonical name is FundamentalTheoremOfCalculusForKurzweilHenstockIntegral.
Accessed 592 times total.

Classification:

AMS MSC:

26A42 (Real functions :: Functions of one variable :: Integrals of Riemann, Stieltjes and Lebesgue type)

Pending Errata and Addenda

None.

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