On modified Takagi functions of two variables
In this paper explicit examples are constructed of continuous functions of two variables whose non-trivial contours have union of Lebesgue measure zero. This follows the work of the reviewer [Z. Wahrscheinlichkeitstheor. Verw. Geb. 52, 267-276 (1980; Zbl 0431.60056)], who established this result for multiparameter Brownian motions [and incidentally the Baire category analogue in Bull. Lond. Math. Soc. 14, 30-32 (1982; Zbl 0485.54023)]. The construction uses a modification of the Takagi function, a beautifully simple early example of a nondifferentiable function. The resulting function is shown to have graph of Hausdorff dimension 2, which is the smallest possible value. Thus the function is much smoother than Brownian motion.
scientific article; zbMATH DE number 3985546