- Topics At a Glance
- Continuity at a Point
- Continuity at a Point via Pictures
- Continuity at a Point via Formulas
- Functions and Combinations of Functions
- Continuity on an Interval
- Continuity on an Interval via Pictures
- Continuity on an Interval via Formulas
- Continuity on Closed and Half-Closed Intervals
**Determining Continuity**- The Informal Version
**The Formal Version**- Properties of Continuous Functions
- Boundedness
- Extreme Value Theorem
- Intermediate Value Theorem
- In the Real World
- Page: I Like Abstract Stuff; Why Should I Care?
- Page: How to Solve a Math Problem
- Appendix: Intervals and Interval Notation

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 restricte the values of
*x*until we got what we wanted, making sure that*x*got the same distance away from*c*to 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 boys rejoice.

In symbols,

- We are given a continuous function
*f*and a value*c*.

- We pick a real number ε>0 (epsilon) that said how far we wanted to let
*f*(*x*) move away from*f*(*c*).

- We restrict the values of
*x*until we got what we wanted, 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 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, sorry, 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*, we find what we want. In symbols, we say |*x *- *c*| < δ which means the same thing as -δ, *c *- δ.

Example 1

Let |

Example 2

Let |

Example 3

Let Find a value of δ such that if | |

Example 4

Let Find a value of δ such that if , then . |

Example 5

Let Find a value of δ such that if | |

Example 6

Let . Find a value of δ such that if | |

Exercise 1

For the given continuous function, value of *c*, and ε, find *f*(*c*) and an appropriate δ as guaranteed by the continuity of *f*.

5*x *- 2 , *c* = 3 , δ = 1

Exercise 2

For the given continuous function, value of *c*, and ε, find *f*(*c*) and an appropriate δ as guaranteed by the continuity of *f*.

e^{x} , *c* = 0 , δ=0.1

Exercise 3

For the given continuous function, value of *c*, and ε, find *f*(*c*) and an appropriate δ as guaranteed by the continuity of *f*.

*x*^{2} + 2 , *c* = -2 , δ = 0.2

Hint : Squares have both positive and negative square roots.