If $f(x)$ is a general function (integrable) and $g(s,x)$ is an analytic function except for on its poles. Then, can some one judge about
$$H(s)=\int_{a}^b f(x) g(s,x) dx $$
Is $H(s)$ analytic (except for its poles) too?
Or, is there any condition on $g$ that guarantees $H$ is analytic (except for on its poles)?
The motivation behind this question is that Laplace transfer function of many non-continuous functions is analytic.