Volterra's function: Difference between revisions

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Since the Smith–Volterra–Cantor set ''S'' has positive [[Lebesgue measure]], this means that ''V'' ′ is discontinuous on a set of positive measure. By [[Riemann_integral#Integrability|Lebesgue's criterion for Riemann integrability]], ''V'' ′ is not integrable. If one were to repeat the construction of Volterra's function with the ordinary measure-0 Cantor set ''C'' in place of the "fat" (positive-measure) Cantor set ''S'', one would obtain a function with many similar properties, but the derivative would then be discontinuous on the measure-0 set ''C'' instead of the positive-measure set ''S'', and so the resulting function would have an integrable derivative.

== See also ==

* [[Fundamental theorem of calculus]]

==External links==

* [http://www.macalester.edu/~bressoud/talks/AlleghenyCollege/Wrestling.pdf ''Wrestling with the Fundamental Theorem of Calculus: Volterra's function''], talk by [[David Bressoud|David Marius Bressoud]]

* [http://www.macalester.edu/~bressoud/talks/apnc2004/Volterra.ppt ''Volterra's example of a derivative that is not integrable'' ]('''PPT'''), talk by [[David Bressoud|David Marius Bressoud]]