This is a question regarding Epsilon-Delta proofs, in this example in Single-Variable Calculus, but hopefully the crux of what I'm asking here, will be general enough.
Introduction
I'm putting this loose introduction here to illustrate the point I'm trying to make about $\epsilon – \delta$ proofs below.
Given a function $f$, it's output $f(x)$ is dependent on its input $x$. Therefore given an arbitrary output $f(x)$ (which we don't know if it exists in the functions Co-Domain) we need to prove that there is a corresponding $x$ that produces it, otherwise the function $f$ would be undefined at that point $x$
However given an $x$ that is in the domain of $f$, we can ALWAYS prove that there exists a $f(x)$, as the Co-Domain of a function is entirely dependent on its Domain.
Onto the $\epsilon – \delta$ Part
The formal definition of a limit (the $\epsilon-\delta$ definition) is :
$$ \lim_{x \to a} f(x) = L \Leftrightarrow \forall \epsilon >0 (\exists \delta>0 \ni (\forall x \in D(0 < |x-a| < \delta \implies |f(x)-L| < \epsilon)))$$
Essentially what this is saying that as we restrict $x$ to being within $\delta$ units of $a$, then as a result of that restriction $f(x)$ becomes restricted to being withing $\epsilon$ units of $L$
You can see this visually below.
Now naturally I would assume that because of this, it would mean that $\epsilon$ is dependent on $\delta$, i.e. as we restrict $x$ to being within $\delta$ units of $a$, then as a result of that restriction $f(x)$ becomes restricted to being withing $\epsilon$ units of $L$. But my assumption is wrong.
A quote from Wikipedia :
Therefore δ depends on ε. The limit statement means that no matter how small ε is made, δ can be made small enough.
Even from the formal definition I referenced it states : $\forall \epsilon > 0, \exists\delta > 0$, implying once again that $\delta$ is dependent on $\epsilon$.
Why is that so? Why does the Epsilon-Delta definition of a limit, treat values of $\epsilon$ which can be elements of a functions Co-Domain, as being independent, and values of $\delta$ which can be elements of a functions Domain as being dependent on $\epsilon$?
Or am I wrong, and does $\epsilon$ and $\delta$ not represent elements of a functions co-domain or domain (i.e. they have nothing to do with the functions domain and co-domain), but rather concepts of distances? With $\epsilon$ being the "error distance", and $\delta$ being the "dependent distance".
Another quote from Wikipedia :
In these terms, the error ($\epsilon$) in the measurement of the value at the limit can be made as small as desired by reducing the distance ($\delta$) to the limit point.
But in this quote it seems that $\epsilon$ is dependent on $\delta$ as it says the error ($\epsilon$) can be made as small as possible, by reducing the distance ($\delta$), implying $\epsilon$ is dependent on $\delta$, so which is it?
To Summarize
I know this is a long post, but I've written all of this out to clarify my thought processes so you can see where I'm coming from.
In a nutshell I am asking two separate questions?
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Are $\epsilon$ and $\delta$ meant to be thought of purely as distances? (which I'm sure they are, otherwise contradictions pop up)
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Is $\epsilon$ dependent on $\delta$, or is $\delta$ dependent on $\epsilon$? i.e. is the "error distance" $\epsilon$ dependent on the "restricting distance" $\delta$ or is it the other way around?
If you have found any gaps, in my knowledge or understanding, please inform me, as an undergraduate student majoring in Pure Mathematics, I'm always looking to improve my understanding. Secondly if a deeper understanding on mathematically rigorous definition of limits, can be found by concepts in Real Analysis, I'd love to hear about them.
Best Answer
Indeed both $\epsilon$ and $\delta$ are distances.
Maurice Fréchet generalized this idea further by defining Metric spaces. For example on $\mathbb R$ the usual metric is called the Euclidean metric where the distance of $x, y \in \mathbb R$ is $|x-y|$.
So $|f(x) -L| < \epsilon$ means the distance of $f(x)$ from $L$ is less than $\epsilon$.
You are correct, by definition $\delta$ depends on $\epsilon$.
A better wording for your second quote in my opinion would be the following:
One can make the error arbitrarily small by choosing a small enough $\delta$.
See also this question.
Hope this helps.