continuous derivative implies bounded variation

Theorem.  If the real function f has continuousMathworldPlanetmath derivative on the interval[a,b],  then on this interval,

Proof.1o¯. The continuous function |f| has its greatest value M on the closed intervalDlmfMathworld[a,b],  i.e.


Let D be an arbitrary partition of  [a,b],  with the points


Consider f on a subinterval[xi-1,xi].  By the mean-value theorem, there exists on this subinterval a point ξi such that  f(xi)-f(xi-1)=f(ξi)(xi-xi-1).  Then we get


Thus the total variation satisfies

supD{all SD’s}M(b-a)<,

whence f is of bounded variation on the interval  [a,b].

2o¯. Define the functionsMathworldPlanetmath G and H by setting


We see that these are non-negative and that  f=G-H.  Define then the functions g and h on  [a,b]  by


Because G and H are non-negative, the functions g and h are monotonically nondecreasing.  We have also


whence  f=g-h.  Since G and H are by their definitions continuous, the monotonic functions g and h have continuous derivatives  g=G,  h=H.  So g and h fulfil the requirements of the theorem.

Remark.  It may be proved that each function of bounded variation is difference of two bounded monotonically increasing functions.

Title continuous derivative implies bounded variation
Canonical name ContinuousDerivativeImpliesBoundedVariation
Date of creation 2013-03-22 17:56:32
Last modified on 2013-03-22 17:56:32
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Theorem
Classification msc 26A45
Related topic ProductAndQuotientOfFunctionsSum