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, we have:

$$ ∀\epsilon ∀x ∃δ(x): |x−x_0|<δ(x)⟹|f(x)−f(x_o)|<ϵ$$

With epsilon, delta being in $\mathbb{R_{\geq 0}}$ and $f: D \to \mathbb{R}$ 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?