Bounded Variation

Author

Eric W. Weisstein
April 14, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/BoundedVariation.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/BoundedVariation.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Additional Author

Text and graphics by
Todd Rowland

Plots of x sin (1/x) and (x^2) sin (1/x)

These superimposed plots show one function of bounded variation, (x^2)sin (1/x) in purple, and one which is not, x sin (1/x).  The former has a continuous first derivative on a bounded domain, hence bounded itself, and the function must have bounded variation.  Alternatively, a direct calculation is possible.  The latter function does not have bounded variation because its partial sums are comparable to the harmonic series.  Namely, ∑ (1/((2n+1/2)π))sin (2n +(1/2))π - (1/((2n+3/2)π))sin(2n+(3/2))π = ∑ (1/(nπ)).

<<Graphics`Colors`;

Plot[{x  Sin[1/x],x^2 Sin[1/x]},{x,0,1},PlotRange->{{-.02,.4},{-.25,.25}},Ticks->False,PlotStyle->{Red,Purple},AxesLabel->TraditionalForm/@{x,"\!\(TraditionalForm\`x\\\\ \(sin(\*SuperscriptBox[\(x\), \(-1\)])\)\), \!\(TraditionalForm\`\*SuperscriptBox[\(x\), \(2\)]\\\\ \(sin(\*SuperscriptBox[\(x\), \(-1\)])\)\)"}]