Is analysis all about epsilons? (Inspired by another question from here)

Question

Yesterday, while I was browsing the new questions, I came across this one How is analysis beautiful? -- confusion from an algebraist, which brought up an interesting question, especially for the people who are just starting out with analysis: are the "annoying" epsilons everywhere in analysis, even in the really advanced stuff? Does an actual analyst work with them on a daily basis or are they useful just for establishing the fundamental results in analysis (i.e. what is covered in university)?
I agree with the person that posted the other question that this may be really interesting for some aspiring maths majors (myself included) and that's why I decided to post this question, especially since I have been thinking about quite similar things too (as it can be seen here What should I study next if I enjoy thinking about problems such as these?).

Answer 1

Suppose one is working with objects which are hard to pin down. You do not know their nature very well, and it seems impossible to understand them perfectly. What do you do next? Well, the next best thing would be to control the behaviour of the object in some way.

A very simple case of this would be: Suppose you want to solve $f(x) = g(x)$ for some functions $f$ and $g$, and suppose such a problem is very difficult. Well, try to show the next thing which is a little weaker: show that for any degree of accuracy want, the difference between $f$ and $g$ can be fit to this degree of accuracy. More precisely, given $\varepsilon >0$ show that we can find $x$ such that $| f (x) - g(x) | < \varepsilon$.

Seems like a worthwhile pursuit, but maybe that's just me...