Let $f(x)= \lfloor x \rfloor \cdot\cos{(a\cdot x)}$ where $x\in \mathbb{R}$. Find for which real constants $a$ function is continuous.
We know function $ \lfloor x \rfloor$ is continuous apart from integers and $\cos x$ is continuous on $\mathbb{R}$ so we want to make $f(x)= \lfloor x \rfloor \cdot\cos{(a\cdot x)}$ continuous at integers
Hint: Suppose you had found such an $a$. Then $f$ is continuous, as is $x\mapsto \cos (ax)$. Thus, $x\mapsto \frac{f(x)}{\cos(ax)}=\lfloor x \rfloor$ is continuous everywhere where $\cos(ax)\neq 0$. Use this to determine $a$ (or show that the condition can't be fulfilled by any $a$!)