Volterra's function: Difference between revisions

Revision as of 22:52, 11 April 2006

Revision as of 05:57, 14 October 2006

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Volterra's function is differentiable everywhere just as ''f''(''x'') (defined above) is. The derivative ''V'' &prime;(''x'') is discontinuous at the endpoints of every interval removed in the construction of the [[Smith-Volterra-Cantor set|SVC]], but the function is differentiable at these points with value 0. Furthermore, in any neighbourhood of such a point there are points where ''V'' &prime;(''x'') takes values 1 and &minus;1. It follows that it is not possible, for every &epsilon; > 0, to find a partition of the real line such that |''V'' &prime;(''x''₂) &minus; ''V'' &prime;(''x''₁)| < &epsilon; on every interval [''x''₁, ''x''₂] of the partition. Therefore, the derivative ''V'' &prime;(''x'') is not Riemann integrable.

A real-valued function is Riemann integrable [[iff]]if and only if it is bounded and continuous almost-everywhere (''i.e.'' everywhere except a set of [[measure theory|measure]] 0). Since ''V'' &prime;(''x'') is bounded, it follows that it must be discontinuous on a set of positive measure, so in particular the derivative of ''V''(''x'') is discontinuous at uncountably many points.

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