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==Further properties==
Volterra's function is differentiable everywhere just as ''f''(''x'') (defined above) is. The derivative ''V'' ′(''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'' ′(''x'') takes values 1 and −1. It follows that it is not possible, for every ε > 0, to find a partition of the real line such that |''V'' ′(''x''<sub>2</sub>) − ''V'' ′(''x''<sub>1</sub>)| < ε on every interval [''x''<sub>1</sub>, ''x''<sub>2</sub>] of the partition. Therefore, the derivative ''V'' ′(''x'') is not Riemann integrable.
A real-valued function is Riemann integrable [[iff]] it is bounded and continuous almost-everywhere (''i.e.'' everywhere except a set of [[measure theory|measure]] 0). Since ''V'' ′(''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.
==External link==
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