Let $U$ be an open connected set. Let $f_n$ be a sequence of continuous functions converging locally uniformly to $f$. If every function $f_n$ has its own primitive on U, show that f has primitive on U.
Im wondering if the U needs to be simply connected.
Sketch of proof:
At each point $z$ in u there is a ball over which $f_n\rightarrow f$ uniformly. Therefore f is continuous over the ball. Also for every triangle $T$ within the ball the: $\int_Tf=lim$ $\int_T f_n = 0$ since every $f_n$ has primitive. Hence, by morera's theorem f is analytic within the ball. This implies f is analytic everywhere on U. Here is where I'm stuck.
Any solutions?
Best Answer
You already proved that the limit function $f$ is holomorphic in $U$. It remains to show that $$ \int_\gamma f(z) \, dz = 0 $$ for every closed path $\gamma $ in $U$, and that is true because it holds for each $f_n$ (and the uniform convergence on the compact set $|\gamma|$).
$U$ needs not be simply connected for this conclusion.