If I have a power series $\displaystyle\sum_{i=0}^{+\infty} {a_iz^i} \in\mathbb{C}[[z]]$ with radius of convergence $r>0$ and I know that the series $\displaystyle\sum_{i=0}^{+\infty} {a_iz^i}$ converges for all $|z|=r$, can I conclude via Abel's theorem (which gives me uniform convergence on every segment joining a point on the circle $|z|=r$ and the origin) that $\displaystyle\sum_{i=0}^{+\infty} {a_iz^i}$ converges uniformly on the subset of $\mathbb{C}$ $|z|\le r$?
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$\begingroup$ The linked question math.stackexchange.com/questions/390407/… asks for counterexamples where the limit function is required to be continuous. $\endgroup$– Jonas MeyerMay 13, 2013 at 14:35
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No.
There exists a power series which converges pointwise on the unit circle, but is discontinuous on the unit circle. If you had uniform convergence on the closed unit disk, you would in particular have continuity on the circle.
There is an example constructed by Sierpinski. You can see it here, p.282, if you can read mathematics in French. Sorry I don't have an English reference. Also note that I've learned this from Julien Melleray's answer on MO.
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$\begingroup$ Ah, discontinuous convergence on the unit circle. Very clever. +1 $\endgroup$– davidlowryduda ♦Apr 7, 2013 at 11:39
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$\begingroup$ @mixedmath Thanks. Sierpinski was clever, for sure. $\endgroup$– JulienApr 7, 2013 at 11:40