If anyone can give me a push-start on the following proof I would appreciate it:

Suppose $\displaystyle g:\mathbb{R}\times [a,b]\rightarrow \mathbb{R}$ is continuous. Show $\displaystyle f:\mathbb{R} \rightarrow \mathbb{R}$ determined by

$\displaystyle f(x)=\int _{a}^{b}g(x,y)dy$ is continuous.

I know that $\displaystyle f'(x)=\int _{a}^{b}g_x(x,y)dy$ but I'm not sure if that helps us here...