Volterra's function

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

• $V(x)$ is differentiable everywhere
• The derivative $V^{\prime }(x)$ 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 $f(x)=x^{2}\sin {(1/x)}$ for $x\neq 0$ 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 $x_{0}$) is determined, extend the function to the right with a constant value of $f(x_{0})$ 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 $f_{1}(x)$, 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 $f_{2}(x)$, $f'(x)$ is then considered on the smaller interval $1/16$ and two translated copies of the resulting function are added to $f_{1}(x)$. 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 $V^{\prime }(x)$ 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.

External Link

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