Let $\displaystyle A\in\mathbb{R}$ and let $\displaystyle f:A\rightarrow\mathbb{R}$.

How do you prove the last two statements of the criteria are equivalent?

2 - There exists $\displaystyle \epsilon_0 > 0$ such that for every $\displaystyle \delta>0$, there exists $\displaystyle x_{\delta},u_{\delta} \in A$ so that $\displaystyle |x_{\delta} - u_{\delta}|<\delta$ and $\displaystyle |f(x_{\delta}) - f(u_{\delta})| \geq \epsilon_0 $

3 - There exists $\displaystyle \epsilon_0 > 0$ and there exist sequences $\displaystyle (x_n),(u_n)$ in A such that $\displaystyle u_n - x_n \rightarrow 0$ as $\displaystyle n \rightarrow \infty$, yet $\displaystyle |f(x_n)-f(u_n)| \geq \epsilon_0$