General constraints for choosing $\delta$ in an $\epsilon$-$\delta$ proof of continuity

Question

Given the formal definition of continuity in $\mathbb{R}$ at a point $c$ (i.e., the $\epsilon$-$\delta$ definition), it seems to me that one may define $\delta$ in terms of the given $\epsilon$ and/or $c$. I am mainly inquiring about using $c$ because I haven't seen it done in examples. But formally it should be allowed, right?

Answer 1

In my experience this use of $a,$ even if $a$ is given numerically and the function is a simple nonlinear polynomial, usually goes something like in this example, where one wants to show $\lim_{x \to 3}x^2=9.$ One starts with $f(x)-a=x^2-9$ and tries to factor out as high a power of $x-a$ as possible, so here $f(x)-a=(x-3)(x+3).$ We know we can make $|x-3|$ small by using a small $\delta.$ But there is the "other factor" $x+3$ whose absolute value must be bounded by something. So the usual trick is to initially impose some extra condition on $\delta$ like here $\delta<1$ works. So if $|x-3|<1$ then $x \in (2,4)$ and so $x+3 \in (5,7)$ and we see our other factor's absolute value is bounded above by $7.$

So, if it happens that $\delta<1$, we know that $|x^2-9|<7|x-3|,$ which we now put less than or equal to $\varepsilon$ and see we want $\delta<\varepsilon/7.$

We must combine this last restriction with the earlier choice $\delta<1,$ and this is often expressed in a formula using the min function as $\delta=\min(1,\varepsilon/7).$