Ahlfors, states that
Theorem: Let $f(z)$ be analytic on the set $R'$ obtained from a rectangle $R$ (interior and on the boundary of the rectangle) by omitting a finite number of interior points $a_i$. If it is true that
$\lim_{z\to a_i}(z-a_i)f(z)=0$ then the integration of $f(z)$ along $R$ is zero.
Some other books use the hypothesis, Let $f(z)$ be analytic on the set $R'$ obtained from a rectangle $R$ by omitting a finite number of interior points $a_i$ and f(z) is continuous on $R$ (inside and on the boundary of the rectangle), then integration of $f(z)$ over the rectangle R is zero.
My question is : "Is it equivalent that "$f(z)$ is continuous at $a_i$ and analytic everywhere expect a_i" and "$\lim_{z\to a_i}(z-a_i)f(z)=0$ and analytic everywhere expect a_i" ?
Yes. Since all $a_i$ are isolated in $R$, those statements equivalent, due to Riemann's theorem on removable singularities.