function of not bounded variation

Example.  We show that the function

f:x {xcosπxwhenx0,0whenx=0,

which is continuousMathworldPlanetmath in the whole , is not of bounded variationMathworldPlanetmath on any interval containing the zero.

Let us take e.g. the interval  [0,a].  Chose a positive integer m such that   1m<a and the partition of the interval with the points   1m,1m+1,1m+2,,1n  into the subintervals [0,1n],[1n,1n-1],,[1m+1,1m],[1m,a].  For each positive integer ν we have (see this (


Thus we see that the total variationMathworldPlanetmath of f in all partitions of  [0,a]  is at least


Since the harmonic series diverges, the above sum increases to as  n.  Accordingly, the total variation must be infinite, and the function f is not of bounded variation on  [0,a].

It is not difficult to justify that f is of bounded variation on any finite interval that does not contain 0.


  • 1 E. Lindelöf: Differentiali- ja integralilasku ja sen sovellutukset III. Toinen osa.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1940).
Title function of not bounded variation
Canonical name FunctionOfNotBoundedVariation
Date of creation 2013-03-22 17:56:29
Last modified on 2013-03-22 17:56:29
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 6
Author pahio (2872)
Entry type Example
Classification msc 26A45
Synonym example of unbounded variation
Synonym function of unbounded variation