Here is the definition of continuity of a function between metric spaces.
Let $(X, d_X)$ and $(Y, d_Y)$ be two metric spaces. A function $f : X \to Y$ is said to be continuous if for every $ \epsilon > 0$ there exists a $\delta > 0$ such that $d_X(x, y) < \delta \implies d_Y(f(x), f(y)) < \epsilon$
Now in most proofs the writer of the proof omits their calculation of $\delta$ and simply say something along the following lines
" Let $\delta = \frac{\epsilon}{c}$, then we have $$d_X(x, y) < \delta \implies ... \implies d_Y(f(x), f(y)) < \epsilon$$"
(From here on in I'll refer to the above generalized quote as the "seen part of a continuity proof")
But for someone trying to write their own proof this doesn't really give much intuition, to the reader it seems like these $\delta$ are chosen almost magically by the proof writer. This is obviously false as the proof writer has done some algebraic manipulations/rough calculations to see why such a $\delta$ can work.
My question thus is, how (generally) does someone writing up such a proof pick a suitable $\delta$? More explicitly what algebraic manipulations (and by this I mean in the most general sense) are done to arrive at something that shows that proof writer what a suitable choice for $\delta$ would be?
I can think of two procedures.
1. Start off with $d_Y(f(x), f(y))$ and end up with $d_X(x, y)$ to find suitable choice for $\delta$
Initially I guessed that to find a suitable choice of $\delta$ one would need to start off with $d_Y(f(x), f(y))$ and do some algebraic manipulations and through some inequalities arrive at $c(d_X(x, y))$ where $c$ is some constant. And from that one could choose $\delta = \frac{\epsilon}{c}$, then for any $\epsilon > 0$ we would automatically have $d_Y(f(x), f(y)) < \epsilon$. The below chain of inequalities shows this procedure from start to finish.
$$d_Y(f(x), f(y)) = ... < ... < c(d_X(x, y)) < c\delta = \epsilon$$
The above is all of the "rough calculations" needed to be done to choose a suitable $\delta$ . Now in the seen part of my proof I could write
"Let $\epsilon > 0$ be given and let $\delta = \frac{\epsilon}{c}$ then we have $$d_X(x, y) < \delta \implies c(d_X(x, y)) < \epsilon \implies d_Y(f(x), f(y)) < \epsilon$$ "
2. Start off with $d_X(x, y)$ and end up with $d_Y(f(x), f(y))$ to find suitable choice for $\delta$
I was told that I should start off with $d_X(x, y)$ and end up with $d_Y(f(x), f(y))$ to find suitable choice for $\delta$. In that case one would need to one would need to start off with $d_X(x, y)$ and do some algebraic manipulations and through some inequalities arrive at $c(d_Y(f(x), f(y)))$ where $c$ is some constant. And then from that choose $\delta = c\epsilon$ and then do a bit more work from that to get $d_Y(f(x), f(y)) < \epsilon$. The below chain of inequalities shows this procedure from start to finish.
$$d_X(x, y) < ... < cd_Y(f(x), f(y)) < c\epsilon$$
which suggests we choose $\delta = c\epsilon$. Then in the seen part of the proof I could write
"Let $\epsilon > 0$ be given and let $\delta = c\epsilon$, then we have $$d_X(x, y) < \delta \implies cd_Y(f(x), f(y)) < \delta \\ \implies cd_Y(f(x), f(y)) < c\epsilon \implies d_Y(f(x), f(y)) < \epsilon$$
In either case I seem to be arriving at the same outcome (correct me if I'm wrong). So my question is what is the difference between those two procedures I outlined above?
Is there any better or more universal way to arrive at a suitable choice for $\delta$? Also is this a one method fit's all kind of thing, i.e. can I do this to prove continuity of any continuous function between metric spaces (I understand that proving some of the inequalities may be very difficult)?
Finally if the first procedure I showed is incorrect, please indicate why it is incorrect (maybe it's the case there that I'm choosing an $\epsilon$ for a specific $\delta$, or maybe the whole chain of inequalities is incorrect, if so please tell me).
Also before this is downvoted (either due to length, or something that may seem obvious), note that this is something that many students may be struggling with, as I don't think I've ever come across an explanation that shows how to systematically choose a suitable $\delta$ (because as I've mentioned before nobody writes up their rough calculations in a proof).
It all depends on the complexity of the problem.
Note that $\delta $ is not always proportional to $\epsilon$ and as you mentioned, it is usually the hidden work which makes it hard to understand the proof.
Let us start with an example.
Prove that $$f(x)=x^2+3x-2$$ is continuous at $x=5.$
Suppose $\epsilon>0 $ is given.
We have to choose a $\delta >0 $ such that $$ |x-5| <\delta \implies |x^2+3x-40|<\epsilon $$
Now we go to our magic corner to find a suitable $\delta $
Note that $$|x^2+3x-40|=|(x-5)(x+8)|$$
We can control $|x-5|$ by our invisible $ \delta$ , so we need to worry about $|x+8|$
We say $x$ is around $5$ so if $\delta <1$ then $$ |x-5|<\delta \implies 4<x<6$$
and as a result $$12 <x+8 <14$$ which implies $$|x+8|<14$$
Now we return to our desk and write, Let $$\delta = min\{1,\frac {\epsilon}{14}\}$$
$$ |x-5|<\delta \implies |x^2+3x-40|= |(x-5)(x+8)|<\delta (14)\le 14(\frac {\epsilon}{14}) =\epsilon $$