# The Formal Version

When we graph continuous functions, three things happen:

• We are given a continuous function f and a value c.
• We decide how far we wanted to let f(x) move away from f(c).
• We restrict the values of x until we get what we want, making sure that
• x is the same distance away from c on either side, and that
• we didn't restrict x to just equal c, since then we would find a dot.

Enter the Greek letters—frat bros rejoice.

In symbols,

• We're given a continuous function f and a value c.
• We pick a real number ε > 0 (epsilon) that specifies how far we want to let f(x) move away from f(c).
• We restrict the values of x until we get what we want, ending with c δ

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 can find our desired value.

We have a great recipe for cooking up δ with the ingredients f, c, and ε. First we combine flour, baking soda, and salt in a bowl, then we...no wait. That's the recipe for Nestle Tollhouse Cookies.

• Write down the inequality f(c) - εf(x) < f(c) + ε and fill in whatever we are given for c, f, and ε.

• Solve the inequality for x.

• Subtract c from all parts of the inequality and find δ.

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 |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 -εf(x) – f(c) < ε which is the same thing as saying f(c) – εf(x) <  f(c) + ε.

In pictures,

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, f(x) is within ε of f(c). In symbols, we say |c| < δ.

But as always, a picture is worth a thousand Greek symbols.
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