- 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

Remember, *f* is continuous on an interval if we can finger paint over *f* on that interval without lifting our drawing digit.

Look at the function *f* drawn below:

- The function
*f*is continuous on the interval (-5,5) because if*c*is any point in (-5,5), . We can start our pencil out on the graph at*x*= -5 and trace the graph to x = 5 without lifting the pencil. - This function is not continuous on the interval (5,8) because
*f*is not continuous at*x*= 7. When*x*= 7 we need to lift the pencil to trace the graph. - Here's a tricky one: the function
*f*is continuous on the interval (5,7). The only point on the whole graph at which*f*is discontinuous is*x*= 7, and 7 isn't in the interval (5,7).

Exercise 1

Look at the function *f* drawn below:

Determine whether the function is continuous on the given interval. If not, state the points in the interval at which *f* is discontinuous.

- (-5,-3)

Exercise 2

Determine whether the function is continuous on the given interval. If not, state the points in the interval at which *f* is discontinuous.

- (-5,3)

Exercise 3

Determine whether the function is continuous on the given interval. If not, state the points in the interval at which *f* is discontinuous.

- (9,10)

Exercise 4

*f* is discontinuous.

- (-3,-1)

Exercise 5

*f* is discontinuous.

- (0,4)