Are power series continuous everywhere where they are defined? Let f(x) ve given by a power series everywhere where ir converges, thus the domain of f(x) consists of the open circle of convergence plus some boundary points of it (posibbly none). Then is f continuous? I know that if f is real valued then yes by abel's limit theorem, but what about complex valued f? I've looked online and found something about a thing called 'Stoltz sector', I didn't really understand it, plus it posibbly doesn't answer my question (therefore I won't quote it).

Ah, I have found the exact answer to your question here: https://mathoverflow.net/questions/110345/does-a-power-series-converging-everywhere-on-its-circle-of-convergence-define-a. The answer provides a 1916 paper from Sierpinski, in which "[he] produces an example where the function converges everywhere on the unit circle but is discontinuous (in fact unbounded) on the circle."
