# Continuity of radius of convergence in complex field [closed]

Let $$f : D \rightarrow C$$ be holomorphic on a disk $$D$$ and let $$R(z_{j})$$ be the radius of convergence of the Taylor expansion of $$f(z)$$ about a point $$z_{j} \in D$$. Show $$R(z)$$ is continuous on $$D$$.

Wlog assume $$f$$ cannot be extended holomorphically on a larger concentric disc $$D_1$$ - otherwise if $$f$$ can be extended to be entire, the radius of convergence is infinity at any point, and if not we use a maximal disc $$D_1$$ where $$f$$ can be extended, and prove the result there hence on $$D$$.
So $$f$$ has some singularities on the boundary of $$D$$ and let $$E$$ the non empty set of them. Since if $$f$$ can be extended holomorphically at a point on the boundary, it obviously is extended on a small arc containing that point, it follows that the complement of $$E$$ is open, hence $$E$$ is closed, hence compact. Then the required radius at a point $$z$$ in $$D$$ is obviously the distance from $$z$$ to $$E$$ (remembering that a holomorphic function on an open disc has a radius of convergence at the center of the disc at least the radius of the disc) and that is a continuos function by elementary topology.