Exercise 8.4.17

Let $F_n(x)={\int }_c^{c+n}f(x,t)\mathit{dt}$ and note that $F_n(x)$ is continuous. So $F(x)=\underset{n\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}{F}_n(x)$ must also be continuous, and since $[a,b]$ is compact $F(x)$ is also uniformly continuous over $[a,b]$.