Equicontinuity of $ f^n (x) = \frac{1}{n} \cos(nx)$

Question

I have to prove the equicontinuity of $ f^{(n)} (x) = \frac{1}{n} \cos(nx)$ and the unequicontinuity of $g^{(n)} (x) = \cos(nx)$ for $f^{(n)}, g^{(n)} := [0,1] -> \mathbb{R}$

I rarely heard anything about equicontinuity and don't know what to do here. Every help is appreciated.

Answer 1

Hint: By the mean value theorem, we have \begin{align} |f^{(n)}(x)-f^{(n)}(y)| \leq |x-y|. \end{align} where the inequality is independent of $n$.