Proof

Consider, for example, the case $I=[a\text{,}\infty )$. Then

$\phi (y)={\displaystyle \underset{b\to \infty}{\mathrm{lim}}}{\int}_{a}^{b}f(x\text{,}y)\mathit{dx}\text{,}$

where the limit is uniform for $y\in [c\text{,}d]$. By Theorem 9.34, for every $b>a\text{,}{\int}_{a}^{b}f(x\text{,}y)\mathit{dx}$ is a continuous function of ...