If $f \in A(D)$, is $f'$ continuous in $\overline D$?

Question

Let $D$ be the open unit disk in $\mathbb C$ and let $A(D)$ be the disk algebra, i.e the algebra of analytic functions in $D$ that are continuous in $\overline D$.

My question is:

If $f \in A(D)$, is $f'$ continuous in $\overline D$?

Answer 1

Not necessarily. Consider $$ f(z)=(1-z)\log(1-z)\quad\text{or}\quad f(z)=\sum_{n=1}^\infty\frac{z^{n!}}{n^2}. $$