Let $F$ be an open family of circles in the complex plane and let $P$ be the union of all the circles from $F$. Suppose that $f$ is a continuous function on $P$ which extends holomorphically from each circle $C$ belonging to $F$ (that is, its restriction to $C$ extends holomorphically through the disc bounded by $C$). Is $f$ holomorphic on $P$? We show that this question is naturally related to a question in $C^2$ and then, for certain families $F$, we show how some standard facts from several complex variables can be used to deal with this question.