- 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

It's good to have a feel for what continuity at a point looks like in pictures. However, sometimes we are asked about the continuity of a function for which we're given a formula, instead of a picture. When this happens, remember that the following three statements must **all** hold for *f* to be continuous at *c*.

- I. The function
*f*is defined at*x*=*c*.

- The limit exists.

- The value
*f*(*c*) agrees with the limit .

Example 1

Determine whether the function
is continuous at |

Example 2

Determine whether the function
is continuous at |

Example 3

Determine whether the function
is continuous at |

Example 4

Determine whether the function is continuous at |

Example 5

At what values is the function |

Exercise 1

Determine whether the function

x = -5 x = -4 x = 0 x = 2 x = 3

Exercise 2

For the function, determine all values at which the function is discontinuous.

Exercise 3

For the function, determine all values at which the function is discontinuous.

Exercise 4

For the function, determine all values at which the function is discontinuous.