You can find your epsilons and deltas in the definition of an open ball. They are just two different ways of writing the same idea. For example, in Definition

You can find your epsilons and deltas in the definition of an open ball. They are just two different ways of writing the same idea. For example, in Definition 2.1, the 'h' in the open ball F is your epsilon and the 'h' in B is the delta. In an epsilon-delta proof, you show that for all |x - c| < delta, |f(x) - f(c)| < epsilon. The open ball B is just the set of all x satisfying |x - c| < delta and the open ball F is the set of all f(x) satisfying |f(x) - f(c)| < epsilon. Any epsilon or delta that you choose implies a set of numbers satisfying those conditions. Those sets are open balls. By using them, you don't have to say things like "all 'x' such that ...". Which method you prefer probably depends on what you're more familiar with and how you tend to think. Open balls can be easier to visualize, if that's how you think. Note that not just any ball will do. Closed balls are open balls that also include their boundary. That is, they use a less-than-or-equal-to instead of less-than. Which you use can make a big difference. An open ball on the real number line is just an open interval, an interval excluding its endpoints. It's easy to generalize: an open ball in a Cartesian plane is a circle excluding its border. In three dimensions, it's a sphere excluding its surface . . . and that's why it's called a ball.