can epsilon in a epsilon-delta proof rely on the independent variable if it cancels out at the end?

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I’m taking a class in Complex Analysis and trying to prove the same result as found in this question.

In the accepted answer, a delta is chosen such that it’s the minimum of two values - one of those values has a $|z_0|$ in it. I was under the impression that delta could not have such variables in it when proving uniform continuity - but the delta is chosen such that the variable is canceled out at the end of the problem.

Is this allowed? I’ve never seen it done before and before I use this on an exam I’d rather know that it’s okay.