Problem: Let $k\in \mathbb{N} $ be fixed. Give an example of two functions $f$ and $g$ neither of which is $C^k$ but the product $f\cdot g$ is $C^k$.
Solution: Consider the functions \[ f(x) = \begin{dcases} 1, &\text{ if } x \in \mathbb{Q} ;\\ 0, &\text{ if } x \notin \mathbb{Q} \end{dcases} \] and \[ g(x) = \begin{dcases} 0, &\text{ if } x \in \mathbb{Q} ;\\ 1, &\text{ if } x \notin \mathbb{Q} . \end{dcases} \] None of the functions is $C^k$, in fact they are not continuous anywhere. But, the product of these two functions are \[ \left( f\cdot g \right) (x) = f(x) \cdot g(x) = 0, \] which is a smooth function.