# A Characterization of Weierstrass Analytic Functions

• Karl Menger
Chapter

## Abstract

The sum theorem and other results of dimension theory make it possible to characterize the graphs of Weierstrass analytic functions among the subsets of C2 (the set of all ordered pairs of complex numbers) without assuming local complex parametrizations and in fact without any reference to plane topology. A complete Weierstrass function F is a set consisting of a power series and all its analytic continuations. Its graph, ϕ(F), is the set of all points (Z0,W0) such that Fincludes at least one power series $${{w}_{0}} + {{a}_{1}}(z - {{z}_{0}}) +\ldots+ {{a}_{n}}{{(z - {{z}_{0}})}^{n}} +\ldots$$ of positive radius of convergence.