Volterra's function

In mathematics, Volterra's function, named for Vito Volterra, is a real-valued function V defined on the real line R with the following curious combination of properties: The function is defined by making use of the Smith–Volterra–Cantor set and an infinite number or "copies" of sections of the function defined by The construction of V begins by determining the largest value of x in the interval [0, 1/8] for which f ′(x) = 0.

Once this is done, a mirror image of the function can be created starting at the point 1/4 and extending downward towards 0.

To construct f2, f ′ is then considered on the smaller interval [0,1/32], truncated at the last place the derivative is zero, extended, and mirrored the same way as before, and two translated copies of the resulting function are added to f1 to produce the function f2.

One can show that f ′(x) = 2x sin(1/x) - cos(1/x) for x ≠ 0, which means that in any neighborhood of zero, there are points where f ′ takes values 1 and −1.

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 a Riemann integrable derivative.