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fundamental theorem of calculus for Kurzweil-Henstock integral
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(Theorem)
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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 $F'(x)$ exists for all $x \in [a,b]$ . Then \begin{equation*} \int_a^b F'(x) dx = F(b)-F(a) . \end{equation*}
The reader should note the subtle difference 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 continuous, and if we use the Lebesgue integral we must assume that $F'$ is Lebesgue integrable. Part of this theorem is that $F'$ 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.
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"fundamental theorem of calculus for Kurzweil-Henstock integral" is owned by jirka.
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Cross-references: interval, unbounded, differentiable, function, necessary, theorem, 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 1134 times total.
Classification:
| AMS MSC: | 26A42 (Real functions :: Functions of one variable :: Integrals of Riemann, Stieltjes and Lebesgue type) |
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Pending Errata and Addenda
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