Volterra's Function - Definition and Construction

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) = x2 sin(1/x) 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 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 will be defined to be 0 outside of the interval . We then translate this function to the interval so that the resulting function, which we call f₁, is nonzero only on the middle interval of the complement of the Smith–Volterra–Cantor set. To construct f₂, f ′ is then considered on the smaller interval, 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 f₁ to produce the function f₂. 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 f₁, f₂, ...