= Continuity of parameter integrals on compact rectangles
{title2=$F(t)=\int_a^bf(t,x)\,dx$}
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.
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