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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.

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    $\begingroup$ How about $z\mapsto\overline{z}$? $\endgroup$ – SmileyCraft Jan 10 at 16:03
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Any function that is not differentiable will do. For instance $f(z)=|z|$.

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