- 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

There are three steps to solving a math problem.

- Figure out what the problem is asking.

- Solve the problem.

- Check the answer.

Example. Determine all values of *x* at which the function

is discontinuous.

Answer:

- Figure out what the problem is asking.

We want to make sure we understand the problem. What does *discontinuous* mean? It means "not continuous", but what does that mean?

A function is continuous at a value *x* = *c* if three things happen:

*f*(*c*) exists,

- exists, and

For the function to be discontinuous at *x* = *c*, one of the three things above need to go wrong. Either

*f*(*c*) is undefined,

- doesn't exist, or

*f*(*c*) and both exist, but they disagree.

This problem is asking us to examine the function *f* and find any places where one (or more) of the things we need for continuity go wrong.

- Solve the problem.

- Where is
*f*(*x*) undefined?

Since we're looking at a rational function, *f* is undefined wherever its denominator is 0. To find where that is, we need to factor the numerator and denominator.

The denominator is 0, therefore *f* is undefined, at *x* = -3 and *x* = 4.

- Where does not exist?

This rational function has a hole at x = -3 and a vertical asymptote at x = 4, therefore doesn't exist. This gives us another reason that *f*(*x*) is discontinuous at x = 4.

- Where do
*f*(*c*) and both exist, but disagree?

This function doesn't have any places like that! Since a rational function is continuous everywhere it's defined, we've found all the discontinuous places we need to worry about.

To summarize, this function is only discontinuous at *x* = -3 and *x* = 4.

- Check the answer.

Besides doing the arithmetic again, probably the best thing to do is graph it on the calculator. Make sure it looks continuous except at *x* = 4, where there should be an asymptote. If we ask the calculator what the function is for *x* = -3, it should say "ERROR," because *f*(-3) is undefined!