Suppose I want to calculate this integral
$$\int_{|z|=r}\frac{f(z)}{z-1}dz$$
for $0<r<1$ and $r>1$, with $f(z)$ an holomorphic function in all $\Bbb{C}$
For the first case ($0<r<1$), I usually use the common argument that $\frac{f(z)}{z-1}$ is holomorphic in $B(0,r)$ for any $r<1$ and that it is continuous in $\partial B(0,r)$. My problem is the following, it is true that $f$ is continuous in $\partial B(0,r)$ for $every$ $r<1$?