Topological genericity of nowhere differentiable functions in the disc and polydisc algebras

Abstract

In this paper we examine functions in the disc algebra A(D) and the polydisc algebra A(DI), where I is a finite or countably infinite set. We prove that, generically, for every f∈A(D) the continuous periodic functions u=Ref|T and u˜=Imf|T are nowhere differentiable on the unit circle T. Afterwards, we generalize this result by proving that, generically, for every f∈A(DI), where I is as above, the continuous periodic functions u=Ref|TI and u˜=Imf|TI have no directional derivatives at any point of TI and every direction v∈RI with ‖v‖∞=1. Finally, we describe how our proofs can be modified to give similar results for nowhere Hölder functions in these algebras.