When we graph continuous functions, three things happen:
Enter the Greek letters, frat boys rejoice.
The continuous function guarantee says that no matter what ε > 0 we pick, we'll be able to find a δ > 0.
We use the Greek letters to specify how close various things are to one another. We use the letter ε to specify how close we want f(x) and f(c) to be. The definition of continuity says for any ε we can find an appropriate δ such that if x is within δ of c, we find our desired value.
The super-formal definition of continuity says:
The function f is continuous at c if for any real ε > 0 there exists a real δ > 0 such that if |x - c| < δ, then |f(x) - f(c)| < ε.
To translate, if f is continuous at c we can pick any real ε > 0 and say we want to have f(x) and f(c) within ε of each other. In symbols, we write this |f(x) - f(c)| < ε.
This is the same thing as saying -ε < f(x) - f(c) < ε which is the same thing as saying f(c) - ε < f(x) < f(c) + ε.
Since f is continuous, we have a guarantee that we can find some real δ > 0 such that if x is within δ of c, we find what we want. In symbols, we say |x - c| < δ which means the same thing as -δ,