I am stuck on a problem about uniform continuity of Lipschitz functions.
A function $F:\mathbb{R}\to\mathbb{R}$ is called Lipschitz continuous if there exists a $L\in\mathbb{R}_{>0}$ such that for all $x,y\in\mathbb{R}$ the inequality $|F(x)-F(y)| \leq L|x-y|$ holds.
(i) Let $F:\mathbb{R}\to\mathbb{R}$ be Lipschitz continuous. Prove the following property (*) for F:
$(*)\qquad \forall \varepsilon>0 \exists \delta>0: \forall x, y \in \mathbb{R}(|x-y|<\delta \Rightarrow F(x)-F(y) \mid<\varepsilon).$
(ii) Let $F:\mathbb{R}\to\mathbb{R}$ be a function that satisfies the condition (*) from (i). Show that F is continuous.
(iii) Find a continuous function $G:\mathbb{R}\to\mathbb{R}$ that does not satisfy the condition (*).
For (i), I think I need to use the definition of Lipschitz continuity and find a suitable δ that depends on ε and L. For (ii), I think I need to use the definition of continuity and show that the condition (*) implies that F is continuous at any point. For (iii), I am not sure how to find a counterexample, but I think it should be a function that oscillates very fast near zero.
I am thankful for any help.