We say that a function has the property of having continuous choice of $\delta$ provided $\epsilon$ for each point in it's domain, if there is an assignment of $\delta$ for each provided $\epsilon$ such that this assignment is sufficient to satisfy the criterion of epsilon delta and further that the assignment regarded as a map is continous.
Example of such a function is $x^2$, we have this following assignment of $\delta$ given an $\epsilon$ that satisfies the criterion:
$$\delta = \min\left(1, \dfrac{\epsilon}{2|x_0| + 1}\right)$$
In the above $\delta(\epsilon)$ is a continuous as a function of $\epsilon$ for a given point $x_0$. My question, does there exist a continuous function for which there can be no continous assignment of $\delta$ for a given input of $\epsilon$? Further would this answer change if we saw $\delta$ rather as a multivariable function of $(x_0,\epsilon)$ and then pose this same but in view of it's multi variable continuity?
Note: No attempt as this question is completly outside of my weightclass. Nonetheless it is a curoisity of mine, and hope someone can show how one could possibly even try answering this.