# Thread: Proof of Non-Uniform Continuity Criteria

1. ## Proof of Non-Uniform Continuity Criteria

Let $A\in\mathbb{R}$ and let $f:A\rightarrow\mathbb{R}$.
How do you prove the last two statements of the criteria are equivalent?
2 - There exists $\epsilon_0 > 0$ such that for every $\delta>0$, there exists $x_{\delta},u_{\delta} \in A$ so that $|x_{\delta} - u_{\delta}|<\delta$ and $|f(x_{\delta}) - f(u_{\delta})| \geq \epsilon_0$

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

2. Assume #2. Then to prove #3 let $\delta=\frac{1}{n}$ for each $n\in \mathbb{Z}^+$.

Assume #3. Then for each $\delta>0$ some term of $\left(x_n-u_n\right)$ must have absolute value less than $\delta$.
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