# Just look carefully! 3

Calculus Level 5

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

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.

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