Is this integral function continuous? Is the following function continuous:
$$
h(t)=\int_{\mathbb{R}^d}f(x)g(t,x)dx
$$
when $t\to g(t,.)$ is continuous  ?
I wanted to prove it by taking a convergent sequence $(t_n)\to t$ and show that $h(t_n)\to h(t)$, but it seems that a limit should be removed inside the integral; so what are the conditions to make such move ? or is there an other way other than using sequences to prove the continuity ? 
Thank you for your time :)

It will depend on the conditions that $f$ and $g$ satisfy. For instance, $h$ will be continuous if


*

*$f$ is integrable

*$g$ is continuous uniformly in $x$, that is, for all $t$ and all $\epsilon>0$ there is a $\delta>0$ (depending on $t$ and $\epsilon$ but not on $x$) such that
$$
|t-s|\le\delta\implies|g(t,x)-g(s,x)|\le\epsilon\quad\forall x\in\mathbb{R}^d.
$$
This follows from the inequality
$$
|h(t)-h(s)|\le\int_{\mathbb{R}^d}|f(x)|\,|g(t,x)-g(s,x)|\,dx.
$$

