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Given a uniformly continuous function $f(x)$ on the real numbers $\Bbb R$, then by the definition of uniform continuity this means: for any $\epsilon>0$ there exists $\delta >0$ such that $|f(x)-f(y)|<\epsilon$ whenever $|x-y|<\delta$. My question is: Given $\epsilon_{n} =\frac{1}{3^{n}}$ (for example), and $y_{n}=x_{n}+w_{n}$ so that $|x_{n} -y_{n}|=|w_{n}|$, then we know that there is $\delta_{n}>0$, how can we find the corresponding $\delta_{n}$? I don't know if my question makes sense.

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    In your first sentence you have written the definition of *uniform continuity*, not of *continuity*.2011-02-28
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    OK, I fixed it.2011-02-28
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    Existence does not imply constructibility. Uniform continuity guarantees that $\delta_n$ exists, it does not guarantee that there is an effective way of *finding it.*2011-02-28
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    What Arturo said is correct, but many functions that are uniformly continuous provide a way to find the corresponding $\delta_n$. Intuitively, there is a maximum first derivative, $L$. Then $\delta_n$ would be $\epsilon_n/L$ but you need to worry about the higher derivatives. Often cutting down $\epsilon$ a little works, as the effect of the higher derivatives falls off faster than the first as $\delta$ goes to 0. If they don't exist, or blow up too badly, you need to work harder.2011-02-28
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    So, in this case there is no way to put conditions on the $w_{n}$'s, i.e. when $|w_{n}|<δ_{n}$ even if we are working on a continuous function $f$ ! Or is there any information that we can get about the $\delta_{n}$, any bounds,...??2011-02-28
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    @Monica: First: you presented a *very* general question, with an arbitrary $f$. My comment is simply that there is no way to give an algorithm in such generality (specific functions *may* have algorithms that tell you how to find $\delta_n$). Second: $\delta_n$ is completely independent of $x_n$, $y_n$, and $w_n$; in the uniformly continuous case, $\delta$ depends *only* on $\epsilon$, so the points are utterly irrelevant. In fact, there is absolutely no point in mentioning them at all, except to clutter up the question.2011-02-28

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