 Continuity of parameter integrals on compact rectangles

If $f$ is jointly continuous on a compact rectangle $J\times [a,b]$, then the displayed parameter integral is continuous on $J$. For $t_n\to t$, pointwise continuity gives $f(t_n,x)\to f(t,x)$. The extreme value theorem bounds $|f|$ by a constant $M$ on the rectangle, and $M$ is integrable on the finite interval $[a,b]$. The dominated convergence theorem gives $F(t_n)\to F(t)$.

The same conclusion holds on an unbounded parameter interval if $f$ is jointly continuous there: use a compact rectangle around each fixed parameter value. A global bound over the whole parameter range is unnecessary.