I am reading a proof of Cauchy's Integral Formula.In the proof,the author let $\phi (z,w)=[f(z)-f(w)]/(z-w)$ if $(z\neq w)$ and $f'(z)$ otherwise and leaves the readers to prove that $g(z)=\phi (z,w)$ is analytic for fixed $w$.I solve this problem by consider the power series around $w$.But I doubt that we can get a more general result.
Let $f:B(a;R) \rightarrow \mathbb{C}$ be continuous and $f$ be analytic on $B(a;R)-\{a\}$,where $B(a;R)$ is a open disc of radius $R$ with centre at $a$.Is it true that $f$ is analytic on $B(a;R)$?