$\fbox{Hypothesis}$
Suppose we have a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ that gives rise to two families of single-variable functions:
$\{f(x,t)$ for fixed $t\in \mathbb{R}\} = \{h_x\}_{x \in \mathbb{R}}$ s.t. $h_x: \mathbb{R} \rightarrow \mathbb{R}$ and is continuous
$\{f(x,t)$ for fixed $x\in \mathbb{R}\} = \{g_t\}_{t \in \mathbb{R}}$ s.t. $g_t: \mathbb{R} \rightarrow \mathbb{R}$ and is Lebesgue integrable
Now suppose further that $\exists g: \mathbb{R} \rightarrow \mathbb{R}$ s.t. $g$ is Lebesgue integrable and $\forall t \in (a,b)$ we have that $|g_t| \le g$.
Now let $F(t) = \int_\mathbb{R} g_t$.
Then I want to show that $F$ is continuous which is true if and only if $\lim_{t_n\to t_0} \int g_{t_n} = \int g_{t_0}$.