Prove that every well defined function satisfies $\epsilon-\delta$ definition if $\exists \delta$ and $\forall x$ quantifiers are swapped [closed]

The motivation for this question is a comment on this post, consider the logical formula:

$$ ∀\epsilon \in \mathbb{R_{\geq 0}}, ∀x \in \mathbb{R}, ∃δ(x) \in \mathbb{R_{\geq 0}}: |x−x_0|<δ(x)⟹|f(x)−f(x_o)|<ϵ$$

With $f: D \to \mathbb{R}$ where $ D \subset \mathbb{R}$.

Now, apparently any function well defined at $x_o$ is supposed to satisfy this. Well, I can see it should satisfy it when $x=x_o$ but $\forall x$ quantifies for all other $x$s in our domain as well. How would I go about proving that?