Just look carefully! 3

$\large \begin{cases} y f(x) + x^nf(y) = f(xy) \\ \displaystyle \lim_{x \to 0} \frac{f(x)}{x} = 1 \end{cases}$

Consider a continuous function $f$ is satisfying the above constraints, where $n$ is any positive integer .

Prove that $f$ is differentiable infinitely many times and $f'$ is continuous for all $n$ .

Find $\displaystyle \sum_{r=1}^n f(\omega^r)$ , where $\omega$ is a primitive $n^\text{th}$ root of unity .