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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 <math>f(x) = x^2 \sin(1/x)</math> for ''x'' ≠ 0 and ''f''(''x'') = 0 for ''x'' = 0. 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 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 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 resulting function, which we call ''f''<sub>1</sub>, is nonzero only on the middle interval of the complement of the Smith–Volterra–Cantor set. To construct ''f''<sub>2</sub>, ''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 ''f''<sub>1</sub> to produce the function ''f''<sub>2</sub>. Volterra's function then results by repeating this procedure for every interval removed in the construction of the Smith–Volterra–Cantor set; in other words, the function ''V'' is the limit of the sequence of functions ''f''<sub>1</sub>, ''f''<sub>2</sub>, ...
==Further properties==
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