After stating the Weierstrass M-test for series of complex functions and the "[itex]f_n[/itex] continuous and uniformly convergeant to f on E ==> f continuous on E" thm, my teacher gives as a corollary that every power series [itex]\sum a_nz^n[/itex] is continuous on its disc of convergence D(0,R). And he doesn't give a proof, as if it's trivial. But I think the corollary is wrong. Am I right in thinking so? The convergence is absolute over all of D(0,R), but we only know for sure that the convergence is only uniform over [itex]\emptyset = \partial D(0,R) \cap U \subsetneq D(0,R)[/itex]. Hence, so is the continuity.