On modified Takagi functions of two variables (Q1086691)

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.