Studying $$f(z)=\int_0^1 g(z,x)\ dx $$
where $g(z,x)$ is analytic in the open unit disk $D$ for all $x\in [0,1]$ and
continuous for $|z| <1$ and $0\leq x \leq1$.Now,
$$\lim_{n\to\infty} [1/n \sum_{k=0}^{n-1} g(z,k/n)]=\int_0^1 g(z,x)\ dx$$ by continuity.
Is the above convergence uniform in $D$?
$D$:=open unit disk