I was trying to prove the equivalence of the $\epsilon$-$\delta$ and topological notions of continuity at a point. (Given the standard topology on a metric space) I could get one direction, but the $\epsilon$-$\delta$ notion of continuity at a point doesn't seem to imply the topological notion of continuity at a point. (I think I might have messed up my definition)
The definition of topological continuity at a point I was using was that a function $f$ is continuous as point $a$ if every open set in the image of $f$ which contain $f(a)$ has an open pre-image.
Basically, the $\epsilon$-$\delta$ notion of continuity at point $a$ only says things about neighborhoods of $a$. But I can always union a neighborhood of $a$ with an open set in some other part of the image to get a new open set. And the $\epsilon$-$\delta$ definition gives me no information about this potentially distant set or its pre-image.
In other words, take $f$ to map some open set $A$ to some open neighborhood $f(A)$ and some closed set $B$ to some open neighborhood $f(B)$ such that $f(A) \cap f(B)=\emptyset$. Also, let it be that f is $\epsilon$-$\delta$ continuous over all of $A$.
So now, let's take some $a \in A$. $f$ is epsilon-delta continuous at $a$. But is it topologically-continuous at $a$? No. Because any open neighborhood of $f(a)$, I can union with $f(B)$ to get an open set whose pre-image is not an open set.
I think my problem is that I got my topological definition of continuity at a point wrong. But I can't figure out how to fix it without invoking concepts from metric spaces.