Mathematical Analysis Fundamentals

(11.33)

converges uniformly for $y\in [c\text{,}d]$, then $\phi \in C(c\text{,}d)$.

Proof

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

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

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 ...