Let $\Omega$ be a Jordan domain in the complex plane whose boundary is piecewise analytic, and let $A(\Omega )$ be the algebra of all holomorphic functions on $\Omega$ that are continuous up to the boundary. We prove the existence of dense linear subspaces and of infinitely generated subalgebras in $A(\Omega )$ all of whose nonzero members are, in a strong sense, not differentiable at almost any point of the boundary. We also obtain infinite-dimensional closed subspaces consisting of functions that are not differentiable at any point of a dense subset of the boundary. In the case of the unit disc, those dense linear subspaces can be found with their functions being nowhere differentiable in the unit circle.
"On Weierstrass' monsters in the disc algebra." Bull. Belg. Math. Soc. Simon Stevin 25 (2) 241 - 262, june 2018. https://doi.org/10.36045/bbms/1530065012