I'm admittedly a bit embarrassed to ask this, but after completing real analysis I came to the realization that I don't actually understand the constraints of continuity as well actually I thought.

Say we're trying to prove $f(x) $ is continuous at $x = x\_0$ using $\epsilon-\delta $. When choosing $\delta$, it is common to express it as a function of $\epsilon$ and $x\_0$, and this part I'm okay with.

However, we *cannot* have it be a function of $x$, and I am struggling to understand why. I can see how it would lead to impossible results if we did, and some part of it is vaguely intuitive. But I have always considered intuition a parlor trick of sorts without having the proper justification behind it, and I'd like some help with clarification.