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Thread: Proof of Non-Uniform Continuity Criteria

  1. #1
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    Proof of Non-Uniform Continuity Criteria

    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 $\displaystyle |f(x_n)-f(u_n)| \geq \epsilon_0$
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  2. #2
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    Assume #2. Then to prove #3 let $\displaystyle \delta=\frac{1}{n}$ for each $\displaystyle n\in \mathbb{Z}^+$.

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