In mathematics, Volterra's function is a real-valued function V(x) defined on the real line R with the following curious set of properties:

• V(x) is differentiable everywhere
• The derivative V′(x) is bounded everywhere
• The derivative is not Riemann-integrable

Definition and construction

The function is defined by making use of the Smith-Volterra-Cantor set and "copies" of the function defined by f(x) = x²sin(1/x) for x ≠ 0 and f(x) = 0 for x = 0. The construction of V(x) begins by determining the largest value of x in the interval [0, 1/8] for which f ′(x) = 0. Once this value (say x₀) is determined, extend the function to the right with a constant value of f(x₀) up to and including the point 1/8. Once this is done, a mirror image of the function can be created starting at the point 1/4 and extending downward towards 0. This function, which we call f₁(x), will be defined to be 0 outside of the interval [0, 1/4]. We then translate this function to the interval [3/8, 5/8] so that the function is nonzero only on the middle interval as removed by the SVC. To construct f₂(x), f ′(x) is then considered on the smaller interval 1/16 and two translated copies of the resulting function are added to f₁(x). Volterra's function then results by repeating this procedure for every interval removed in the construction of the SVC.

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 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′(x₂) - V′(x₁)| < ε on every interval [x₁, x₂] of the partition. Therefore, the derivative V′(x) is not Riemann integrable.

External link

• http://www.macalester.edu/~bressoud/talks/Volterra-4.pdf