fundamental theorem of calculus for Riemann integration

- Let f be a Riemann integrablePlanetmathPlanetmath function on an interval [a,b] and F defined in [a,b] by F(x)=axf(t)𝑑t+k, where k is a constant. Then F is continuousMathworldPlanetmathPlanetmath in [a,b] and F=f almost everywhere (

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

0.1 Observations

Notice that the second fundamental theorem is not a converseMathworldPlanetmath of the first. In the first we conclude that F=f except in a set of measure zeroMathworldPlanetmath (, while in the second we assume that F=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 staircaseMathworldPlanetmath function, defined on [0,1]. We have that

  • F is continuous in [0,1],

  • F=0 except in a set of (this set must be contained in the Cantor setMathworldPlanetmath),

  • f:=0 is clearly a Riemann integrable function and 0x0𝑑t=0.

Thus, F(x)0xF(t)𝑑t.

This leads to the question: what kind functions F can be expressed as F(x)=F(a)+axg(t)𝑑t, for some function g ? The answer to this question lies in the concept of absolute continuity ( (a which the devil staircase does not possess), but for that a more general of integration must be developed (the Lebesgue integration (

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 calculusMathworldPlanetmath (Riemann integral)
Defines second fundamental theorem of calculus (Riemann integral)