I am studying Cauchy's integral theorem from shaum's outline,the theorem states that
Let $f(z)$ be analytic in a region $R$ and on its boundary $C$. Then $$\oint_{C}f(z)dz=0$$
After the statement of this theorem,it is written there:
This fundamental theorem, often called Cauchy’s integral theorem or simply Cauchy’s theorem, is valid for both simply and multiple-connected regions. It was first proved by use of Green’s theorem with the added restriction that $f'(z)$ be continuous in $R$. However, Goursat gave a proof which removed this restriction. For this reason, the theorem is sometimes called the Cauchy –Goursat theorem when one desires to emphasize the removal of this restriction.
My Question:If a function is analytic in a region then we know that all its derivatives are analytic in that region and hence they are continuous,then why this added restriction of continuity was required while proving the theorem with the help of Green's theorem?Whether this fact was not known at that time(the fact that if a function is analytic then it is smooth) and Goursat used it in his proof?