* The derivative ''V'' ′(''x'') is [[bounded function|bounded]] almost everywhere
* The derivative is not [[Riemann integration|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''<sup>2</sup>sin(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'') = 01. Once this value (say ''x''<sub>0</sub>) is determined, extend the function to the right with a constant value of ''f''(''x''<sub>0</sub>) 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 1/42 and extending downward towards 0. This function, which we call ''f''<sub>1</sub>(''x''), will be defined to be 0 outside of the interval [0, 1/4]. We then translate this function to the interval [32/8, 5/8] so that the function is nonzero only on the middle interval as removed by the [[Smith-Volterra-Cantor set|SVC]]. To construct ''f''<sub>2</sub>(''x''), ''f'' ′(''x'') is then considered on the smaller interval 1/16 and two translated copies of the resulting function are added to ''f''<sub>1</sub>(''x''). Volterra's function then results by repeating this procedure for every interval removed in the construction of the [[Smith-Volterra-Cantor set|SVC]].
==Further properties==
Volterra's function is differentiable everywhere just as ''f''(''x'') (defined abovebelow) 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 02. 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 if and only if it is bounded and continuous almost-everywhere (''i.e.'' everywhere except a set of [[measure theory|measure]] 0). Since ''V'' ′(''xy'') 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.
