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When we apply the $\epsilon -\delta$ definition of continuity of function, we normally assumed that $\epsilon$ to be small enough. However, how small should the $\epsilon$ be? Intuitively it is quite reasonable to just mention small imprecisely but if the function vibrate frequently at some point, we need to consider the $\epsilon$ really small but then i don't know how small should it be defined. What do the definition mean by small $\epsilon$?

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I don't assume anything on $\epsilon$ other than the fact that $\epsilon>0$. – Nameless Dec 10 '12 at 16:04
If you're wanting specific values, we usually choose the $\varepsilon$, and proceed to find out what $\delta$ ensures that our function stays with the given $\varepsilon$. It sounds like your asking about the other direction. – Clayton Dec 10 '12 at 16:04
@Clayton ya but the $\epsilon$ you choose can't be too big – Mathematics Dec 10 '12 at 16:05
if $\delta$ works for $\epsilon=10^{-100}$, it certainly works for $\epsilon'=10^{100}$. – Olivier Bégassat Dec 10 '12 at 16:08
@Mathematics: Yes, any positive $\epsilon$... – Thomas Dec 10 '12 at 16:11
up vote 9 down vote accepted

If you look at the definition it actually doesn't say small (or at least it shouldn't since small is a relative term). All you have is that for any $\epsilon >0$ there is a $\delta > 0$ ... So the definition simply just says that no matter what positive $\epsilon$ you can come up with, there has to exist this positive $\delta$ such that ...

Why might someone then think of $\epsilon$ as begin small? That is because the "problem" in finding a suitable $\delta$ becomes "harder" the smaller $\epsilon$ is. There are (in a certain sense) fewer choices for $\delta$ the smaller $\epsilon$ is.

As mention in the comment below (@Daid Mitra) another way to think of why $\epsilon$ should be small is that for small $\epsilon$ we are closer to the point that we are considering.

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In the definition of continuity, there is no mentioning of "small $\epsilon$". It starts with "for every $\epsilon>0$", there is a $\delta>0$...". Intuitively, we are trying to show the existence of a $\delta$ no matter how small $\epsilon$ is.

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The definitions says that for all positive $\epsilon$ you can find some positive $\delta$. In that sense, $\epsilon$ can be considered arbitrarily small.

What you probably think of is the following: Sometimes, in order to chose a suitable $\delta$ we need that $\epsilon$ is smaller than some (positive!) number. This does not hurt insofar as we may always assume without loss of generality that $\epsilon$ is smaller than this number.

For example, if you have an expression for $\delta$ (depending on $\epsilon$) and this works only (e.g. is positive only) if $\epsilon<10$, then you may write in your proof "Wlog., $\epsilon<10$". Indeed, If you thus manage to show that for all $\epsilon>0$ with $\epsilon<10$ there exists a $\delta$ such that ..., then a closer look at the definition of continuity will reveal that this indeed shows that there is such $\delta$ for all $\epsilon>0$: You are "allowed" to use a $\delta$ for a bigger $\epsilon$, even if that $\delta$ is "intended" for some smaller $\epsilon$.

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In the context of $\epsilon$-$\delta$ proofs, the size of $\epsilon$ does not matter; it could be anything greater than zero (by definition).

What you are trying to find is a relationship between $\epsilon$ and $\delta$, and whether it holds for all $\epsilon>0$.

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In addition to the above, it's often the case when using continuity (and other sorts of similar statements) that we'll utilize the $\epsilon-\delta$ definition by picking a very small $\epsilon$ in order to approximate some other quantity. In this case, the smallness of $\epsilon$ depends on the context, and we often leave just how small $\epsilon$ will be until the end of the argument so we can see what will be required.

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The way I understand the $\epsilon -\delta$ relationship is that it's an 'if ... then ...' statement. If you are accurate within $\delta$ distance from your target input value, then you will be within $\epsilon$ distance of your target output value. Certainly, there is no penalty for being more accurate in the input value than you need to be, you will still hit your target output value. The proof of continuity usually involves showing that for the right type of function, the continuous function, there is always a way (at least theoretically) to be precise enough to hit your target output value value, if you get close enough to the target input value.

This can be a real world issue because quite often it's impossible to achieve perfection. In real life, you are never going to perfectly hit the target voltage, tension or what have you. So generally, if you are trying to control an output from a machine or some complicated system, it's nice to know that if you get close enough to the right settings, the result will be close to the way you want. So continuity is a nice property to have.

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The definition states for any epsilon greater than zero, there exists an appropriate delta that satisfies a certain condition. In general, the smaller you choose the epsilon value, the smaller the delta value becomes. We call this an epsilon neighborhood of y, or a delta neighborhood of x. Then no matter how small we choose the epsilon neighborhood (interval around y, which we choose first), we can choose a delta neighborhood (interval around x) so that every x value in that neighborhood gets mapped into the epsilon neighborhood. The smaller the epsilon value is, the smaller (in general) the delta neighborhood must be in order to satisfy the requirement.

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