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 $F(x)={\int}_{a}^{x}f(t)\mathit{d}t+k$, 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 $F(x)F(a)={\int}_{a}^{x}f(t)\mathit{d}t$.
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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 ${\int}_{0}^{x}0\mathit{d}t=0$.
Thus, $F(x)\ne {\int}_{0}^{x}{F}^{\prime}(t)\mathit{d}t$.
This leads to the question: what kind functions $F$ can be expressed as $F(x)=F(a)+{\int}_{a}^{x}g(t)\mathit{d}t$, 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  20130322 17:57:32 
Last modified on  20130322 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) 