integral of limit function

Theorem.  If a sequence f1,f2, of real functions, continuousMathworldPlanetmath on the interval[a,b],  converges uniformly on this interval to the limit functionMathworldPlanetmath f, then

abf(x)𝑑x=limnabfn(x)𝑑x. (1)

Proof.  Let  ε>0.  The uniform continuity implies the existence of a positive integer nε such that

|fn(x)-f(x)|<εb-ax[a,b]  whenn>nε.

The function f is continuous (see and thus Riemann integrablePlanetmathPlanetmath ( (see on the interval.  Utilising the estimation theorem of integral, we obtain


as soon as  n>nε.  Consequently, (1) is true.

Remark 1.  The equation (1) may be written in the form

ablimnfn(x)dx=limnabfn(x)𝑑x, (2)

i.e. under the assumptions of the theorem, the integration and the limit process can be interchanged.

Remark 2.  Considering the partial sums of a series n=1fn(x) with continuous terms and converging uniformly on  [a,b],  one gets from the theorem the result analogous to (2):

abn=1fn(x)dx=n=1abfn(x)𝑑x. (3)
Title integral of limit function
Canonical name IntegralOfLimitFunction
Date of creation 2013-03-22 19:01:41
Last modified on 2013-03-22 19:01:41
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Theorem
Classification msc 26A15
Classification msc 40A30
Related topic TermwiseDifferentiation