Consider a complex power series $\sum a\_n z^n \in \mathbb C[[z]]$ with radius of convergence $0\lt r\lt\infty$ and suppose that for every $w$ with $\mid w\mid =r$ the series $\sum a\_n w^n $ converges .  

We thus obtain a complex-valued function $f$ defined on the closed disk $\mid z\mid \leq r$ by the formula $f(z)=\sum a\_n z^n$. 


**My question:** is $f$ continuous ? 


This is a naïve question which looks like it should be answered in any book on complex analysis.  

But I checked quite a few books, among which the great treatises : Behnke-Sommer, Berenstein-Gay, Knopp, Krantz, Lang, Remmert, Rudin, Stein-Shakarchi, Titchmarsh, ... .  

I couldn't find the answer, and yet I feel confident that it was known in the beginning of the twentieth century. 


**Edit**  

Many thanks to Julien who has answered my question: Sierpinski proved (in 1916) that there exists such a power series $\sum a\_n z^n $ with radius of convergence $r=1$ and associated function $f(z)=\sum a\_n z^n $ not bounded on the closed unit disk and thus certainly not continuous. 


It is strange that not a single book on complex functions seems to ever have mentioned this example. 


On the negative side, I must confess that I don't understand Sierpinski's article at all!  

 He airdrops a very complicated, weird-looking power series and proves that it has the required properties in a sequence of elementary but completely obscure computations.  

I would be very grateful to anybody who would write a new answer with a few lines of explanation as to what Sierpinski is actually doing.



Does a power series converging everywhere on its circle of convergence define a continuous function?
I searched all over for an answer to this question back in my student days. I found the answer in a paper by Sierpinski, "Sur une série potentielle qui, étant convergente en tout point de son cercle de convergence,représente sur ce cercle une fonction discontinue ", which is featured in his collected works, [see here](http://www.plouffe.fr/simon/math/Sierpinski%20Oeuvres%20Choisies%20I.pdf), p282) and apparently was published in 1916.


It does confirm your expectation that this was known in the beginning of the twentieth century (I don't know whether it's the first proof or not, but from the paper it's clear that Sierpinski thought the result to be new).


EDIT: I just realized that not everybody speaks French ;-) so, to be clear: Sierpinski produces an example where the function converges everywhere on the unit circle but is discontinuous on the circle.


