TITLE: Holomorphic function on $D\subset \mathbb{C}$ has to be $\mathcal{C}^\infty (D)$?
QUESTION [0 upvotes]: I'm confused about this, in my notes I have the following:

Theorem: Let $f$ be holomorphic inside and on the boundary ($C$, itself a contour) of a simply connected region $D$. Then $\oint_Cf(z)dz=0$.
Proof:
Assuming that $f'$ is continuous in $D$ and on $C$,...

Now I always thought that a holomorphic function on $D$ is $\mathcal{C}^\infty(D)$. Is that not the case?

REPLY [1 votes]: A standard assumption in this context is that $f$ is holomorphic in $D$ (which is open, so doesn't include the boundary) and continuous on its closure $\overline{D}$.  It doesn't have to be differentiable on the boundary.
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