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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.

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    $\begingroup$ 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. $\endgroup$ – Brevan Ellefsen Nov 29 '18 at 8:24
  • $\begingroup$ Thanks for your reply. How would you do that? $\endgroup$ – M. Rahmat Nov 30 '18 at 4:47
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    $\begingroup$ 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 $|z|$) $\endgroup$ – Brevan Ellefsen Nov 30 '18 at 7:16
  • $\begingroup$ Thanks. I will try... $\endgroup$ – M. Rahmat Nov 30 '18 at 19:37
  • $\begingroup$ Success I hope? $\endgroup$ – Brevan Ellefsen Dec 3 '18 at 16:45

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