Being that it is the end of the semester I have an Analysis final coming up in a few days. On previous tests my professor has asked for a proof that a certain function was continuous explicitly requiring that the proof use the $\varepsilon$-$\delta$ method of continuity.
The thing is, like many students, I really don't like the $\varepsilon$-$\delta$ method. I much prefer to prove that the pre-image of every open set on the domain is open, I find it more intuitive and it was what I was using for a while before questions requiring the $\varepsilon$-$\delta$ method started popping up.
In a proof by example, Hagen von Eitzen pointed out that there are certainly many cases where it is a better idea to use $\varepsilon$-$\delta$ proofs over the topological ones. However there are still cases where I find it easier to the topological definition.
My question is it there some way I could convert a proof using the pre-image of open sets definition to a $\varepsilon$-$\delta$ format? Or use my intuitions of one proof method to generate proofs in the other? My main issue with rewriting my proofs is that the pre-image definition deals with continuity of an interval rather than the continuity of a single point, while the $\varepsilon$-$\delta$ definiton defines continuity at a single point.
I understand that $\varepsilon$-$\delta$ proofs are an important skill to have, and that my professor probably has good intentions in requiring them. I plan to practice my $\varepsilon$-$\delta$ skills before the test, but I still feel nervous I might blank on the test and I would like to have a back-up plan for these types of questions.