I am not being able to solve this problem of Continuity.

Question

Prove that $[x] \sin^2(\pi x)$ is continuous at every integer point and $[x] \cos^2 (\pi x)$ is discontinuous at every integer point.

Answer 1

Hint: $\sin(\pi x) = 0$ whenever $x\in\mathbb{Z}$ and $|\cos(\pi x)| = 1 \not= 0$ whenever $x \in \mathbb{Z}.$