Chapter 1 - Limits and Their Properties - 1.4 Exercises - Page 80: 42

The function is continuous over the set of real numbers and has no discontinuities.

Using Theorem $1.12:$ Since both $f(x)=\dfrac{\pi x}{2}$ and $g(x)=\cos{x}$ are continuous over the set of real numbers; then $g(f(x))=\cos{\dfrac{\pi x}{2}}$ is continuous over the set of real numbers.