fundamental theorem of calculus for Riemann integration

In this entry we discuss the fundamental theorems of calculus for Riemann integration.

- Let $f$ be a Riemann integrable function on an interval $[a,b]$ and $F$ defined in $[a,b]$ by $F(x)=\int_{a}^{x}f(t)\,dt+k$ , where $k\in\mathbb{R}$ is a constant. Then $F$ is continuous in $[a,b]$ and $F^{\prime}=f$ almost everywhere (http://planetmath.org/MeasureZeroInMathbbRn).

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- Let $F$ be a continuous function in an interval $[a,b]$ and $f$ a Riemann integrable function such that $F^{\prime}(x)=f(x)$ except at most in a finite number of points $x$ . Then $F(x)-F(a)=\int_{a}^{x}f(t)\,dt$ .

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0.1 Observations

Notice that the second fundamental theorem is not a converse of the first. In the first we conclude that $F^{\prime}=f$ except in a set of measure zero (http://planetmath.org/MeasureZeroInMathbbRn), while in the second we assume that $F^{\prime}=f$ except in a finite number of points. In fact, the two theorems can never be the converse of each other as the following example shows:

Example : Let $F$ be the devil staircase function, defined on $[0,1]$ . We have that

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$F$ is continuous in $[0,1]$ ,
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$F^{\prime}=0$ except in a set of (this set must be contained in the Cantor set),
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$f:=0$ is clearly a Riemann integrable function and $\int_{0}^{x}0\,dt=0$ .

Thus, $F(x)\neq\int_{0}^{x}F^{\prime}(t)\,dt$ .

This leads to the question: what kind functions $F$ can be expressed as $F(x)=F(a)+\int_{a}^{x}g(t)\,dt$ , for some function $g$ ? The answer to this question lies in the concept of absolute continuity (http://planetmath.org/AbsolutelyContinuousFunction2) (a which the devil staircase does not possess), but for that a more general of integration must be developed (the Lebesgue integration (http://planetmath.org/Integral2)).

Title	fundamental theorem of calculus for Riemann integration
Canonical name	FundamentalTheoremOfCalculusForRiemannIntegration
Date of creation	2013-03-22 17:57:32
Last modified on	2013-03-22 17:57:32
Owner	asteroid (17536)
Last modified by	asteroid (17536)
Numerical id	10
Author	asteroid (17536)
Entry type	Theorem
Classification	msc 26A42
Related topic	FundamentalTheoremOfCalculusClassicalVersion
Related topic	FundamentalTheoremOfCalculus
Defines	first fundamental theorem of calculus (Riemann integral)
Defines	second fundamental theorem of calculus (Riemann integral)