absolute convergence of integral and boundedness of derivative

Theorem.  Assume that we have an http://planetmath.org/node/11865absolutely converging integralDlmfPlanetmath


where the real function f and its derivative f are continuousMathworldPlanetmath and f additionally bounded on the interval[a,).  Then

limxf(x)= 0. (1)

Proof.  If  c>a,  we obtain


from which

(f(c))2=(f(a))2+2acf(x)f(x)𝑑x. (2)

Using the boundedness of f and the absolute convergenceMathworldPlanetmath, we can estimate upwards the integral


whence a|f(x)f(x)|𝑑x is finite and thus af(x)f(x)𝑑x converges absolutely.  Hence (2) implies


i.e. limx(f(x))2 exists as finite, therefore also


Antithesis:  A>0.  It implies that there is an x0(a) such that


If now  b>x0, then we had


This means that x0|f(x)|𝑑x and consequently also a|f(x)|𝑑x would be divergent.  Since it is not true, we infer that  A=0,  i.e. that the assertion (1) is true.

Title absolute convergence of integral and boundedness of derivative
Canonical name AbsoluteConvergenceOfIntegralAndBoundednessOfDerivative
Date of creation 2013-03-22 19:01:28
Last modified on 2013-03-22 19:01:28
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Theorem
Classification msc 40A10
Related topic NecessaryConditionOfConvergence