# 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*) -*ε*<*f*(*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 |*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*.

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 |*x *- *c*| <* δ.*