Prove that f is a lipschitz function if $\exists K \in \mathbb{R^+} \forall x,y \in \mathbb{R}: |f(y)-f(x)| \le K|\cos y - \cos x|$

Question

I'm trying to prove that $f : \mathbb{R} \rightarrow \mathbb{R} \phantom{2}$ is a Lipschitz function if:

$\exists K \in \mathbb{R^+} \phantom{1}\forall x,y \in \mathbb{R}: \lvert f(y)-f(x) \rvert \le K\lvert \cos y - \cos x \rvert$

It seems easy because it is so similar to the definition of a lipschitz function but I'm not sure how can I prove it, any suggestions?

Answer 1

Hint: We know by mean value theorem $\frac{|\cos y - \cos x|}{|y-x|}\le 1 \implies |\cos y - \cos x|\le|x-y|$ $\forall x,y \in \mathbb{R} $ as $\cos$ is differentiable over all of $\mathbb R$.

So you have $$\exists K \in \mathbb{R^+} \forall x,y \in \mathbb{R}: |f(y)-f(x)| \le K|\cos y - \cos x|\le K|x-y|$$