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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