Example of a Continuous Nowhere-Differentiable Function with the Modulus of Continuity not Exceeding a Given Value

Abstract

For an arbitrary convex non-Lipchitz modulus of continuity ω(t), we construct a continuous nowhere- differentiable function ϕω(x), whose modulus of continuity does not exceed ω(t) and that has zero derivative number at every point, is constructed. This construction follows the work of B. Bolzano for the continuous nowhere-differentiable function. The function fω(z) = fω(x + iy) := ϕω(x) is a continuous nowheredifferentiable function, even if it is considered as a function of two real variables, whose modulus of continuity does not exceed ω(t) and that has zero derivative number at every point along two noncollinear directions. A sufficient condition of analyticity is obtained in this work under the assumtption that the function satisfies of the Lipschitz condition at every point ζ along some set Eζ rather than the conventional assumption that the function has a derivative with respect to z at every point ζ along some set Eζ. Such a function fω(z) shows that the former assumption cannot be weakened in this theorem.