Regarding continuous function not in the Disc algebra

Question

Let $D=\{z\in\mathbb{C}: |z|<1\}$.

$C(\bar{D})=\{f:\bar D\longrightarrow \mathbb{C}: f \;\text{is continuous on}\; \bar{D}\}$

$A(D)=\{f\in C(\bar{D}): f \;\text{is analytic in} \;D\}$ Can you give me example of a function which is in $C(\bar{D})$ but not in $A(D)$? A function that is not analytic exactly at the boundary but continuous.

Answer 1

The issue is not the boundary, as analiticy is only considered on open sets. What you want is a function that is continuous but not analytic. So any function that is continuous but not differentiable will do. For instance $f(z)=|z|$.