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

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