Continuity of hypergeometric function in the arguments

Question

For $|z| < 1$, is the Gaussian hypergeometric function $_{2}F_1(a,b;c;z)$ continuous in the arguments $a$, $b$, and $c$? How to prove it?

Answer 1

For this hypergeometric function to exist at all, we require that $c$ is not a nonpositive integer ($c \notin -\mathbb N$). Then using the Ratio Test, the series converges absolutely for $|z| < 1$, and convergence is uniform on any compact subset of $\{(a,b,c,z) \in \mathbb C^4: c \notin -\mathbb N, |z| < 1\}$. A uniform limit of continuous functions on a compact set is continuous.