# The infimum of the radii of convergence of a real analytic function over a compact

Suppose $$f$$ is real analytic on a domain $$D$$ of $$\mathbb{R}^n$$ with $$n\geq1$$. By definition at each point $$x$$ of $$D$$ the Taylor series expansion of $$f$$ converges; let $$r(x)$$ be the radius of convergence of this series. Again, by definition, $$r(x)$$ is never zero as $$x$$ runs over $$D$$. Now, let $$K$$ be a compact of $$D$$. Prove that the infimum of these $$r(x)$$, as $$x$$ runs over $$K$$ is strictly positive.

• It's a compact set. Can you show $r(x)$ is continuous? Continuous functions achieve their extrema on compact sets, and the minimum is greater than zero so the infimum = minimum is as well. – Brevan Ellefsen Nov 29 '18 at 8:24
• Thanks for your reply. How would you do that? – M. Rahmat Nov 30 '18 at 4:47
• Depends on what tools you have. Given you have a complex analysis tag: The radius of convergence is simply the distance to the nearest singularity. The distance function is necessarily continuous (e.g., since $\mathbb C$ has norm $|z|$) – Brevan Ellefsen Nov 30 '18 at 7:16
• Thanks. I will try... – M. Rahmat Nov 30 '18 at 19:37
• Success I hope? – Brevan Ellefsen Dec 3 '18 at 16:45