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==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'') = 0. 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 1/4 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 [3/8, 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==
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