Answer
The function is continuous everywhere except at the nonremovable discontinuities$$x= \{ 2n+1 \mid n \in \mathbb{Z} \}.$$
Work Step by Step
As we know, the trigonometric functions $\sin x$ and $\cos x$ are continuous everywhere, so the function $f(x)= \tan \frac{ \pi x}{2}=\frac{ \sin \frac{ \pi x}{2}}{\cos \frac{ \pi x}{2}}$ is continuous everywhere except at those points vanishing the denominator. So the discontinuities of the function are$$\cos \frac{\pi x}{2}=0 \quad \Rightarrow \quad x= \{ 2n+1 \mid n \in \mathbb{Z} \}$$These discontinuities are nonremovable since as we approach these points, $\tan \frac{ \pi x}{2}$ tends to infinity, so we cannot redefine the function at these points.
