Let $\displaystyle J_0(x) =2/\pi \int_{0}^{\pi/2}{\cos(x \cos{t})}dt $

I need to show that $\displaystyle J_0$ is a continuous bounded function. The fact that it is bounded seems completely obvious since cos is bounded. The continuity is causing some problems however. Clearly, cos is continuous and the process of "integrating" is continuous, but this doesn't give a proof.