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Prove or refute (by means of a counterexample) the following claim:

Let $\,f\,$ be an analytic function in $\,D:=\{z\;\;:\;\;|z-a|<r\,\}\,$ and continuous in $\,\overline{D}\,$ , then: there exists $\,\delta>0\,$ s.t. $\,f\,$ can be analitically continued to $\,\{z\;\;:\;\;|z-a|<r+\delta\}\,$

What I tried: At first I though this can be disproved by means of the example $$\frac{1}{1-z}=\sum_{n=0}^\infty z^n\,\,,\,|z|<1$$ but then I realized that it can't be said the function $\,\displaystyle{\frac{1}{1-z}}\,$ is continuous on $\,\overline D\,$ , so I began suspecting the claim is true.

We can assume $\,a=0\,$ to avoid cumbersome stuff.

Now, we have that $$\forall\,z\in D\,\,,\,\,f(z)=\sum_{n=0}a_nz^n$$ so I argued: suppose $\,z_0\in \partial D\,$ is s.t. $\,f\,$ cannot be continued through it. Let us choose a sequence $\,\{z_k\}\subset D\,\,\,s.t.\,\,\,z_k\xrightarrow [k\to\infty]{}z_0\,$ , then by continuity we get $$f(z_0)=\lim_{n\to\infty}f(z_k)=\lim_{k\to\infty}\sum_{n=0}^\infty a_nz_k^n\stackrel{\color{red}{attention!}}=\sum_{n=0}^\infty a_nz_0^n$$

Of course, the above is possible if we assume we can interchange the limit and the summation, which I'm not sure at all...I can't see here any monotone or dominated convergence theorem that'll allow this, so I'm pretty stuck.

Any hint, idea or solution will be duly appreciated.

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Abel's Theorem seems relevant. – Ragib Zaman Jul 1 '12 at 16:00
up vote 3 down vote accepted

Since the hints that were posted in the comments seem to have disappeared, I'll give a solution here.

Let $$f(z) = \sum_{n=1}^\infty \frac{z^n}{n^p}$$ where $p>1$. Then the radius of convergence is 1 and the series converges unformly (hence to a continuous function) on the closed unit disc.

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This is weird: AFAIK, the only way an answer together with the comments after it can disappear is if the answer's poster deletes this what happened here? – DonAntonio Jul 2 '12 at 12:40
There were hints giving (more or less) the above solution, but I guess the poster deleted them for some reason. – mrf Jul 2 '12 at 13:05

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