The document explains the concept of continuity in functions, noting that continuous functions can be represented without lifting the pencil. It discusses how to determine if a given function is continuous at specific points and introduces types of discontinuities, including removable and essential discontinuities. Examples are provided to illustrate these concepts.

1. Continuity.
http://www.calculus-help.com/continuity/

2. Continuous functions can be
drawn with a single unbroken
pencil stroke.

3. Continuous functions can be
drawn with a single unbroken
pencil stroke.

4. Is the function shown continuous?

5. Is the function shown continuous?

6. Is the function shown continuous?

7. Is the function shown continuous?

8. Is the function shown continuous?

9. Examples.
Use the definition of continuity to determine whether or not the function
2
f(x) = 3x + x - 4 is continuous for x = 2.

10. Examples.
Sketch the graph of the function below and give the location of any
discontinuities.

11. Opener.
1. Can we use f(c) to evaluate if we know that f is a continuous
function?
2. Name one type of function for which always.
3. For what values of c can we use f(c) to evaluate if
is a rational function?

12. Removable discontinuity.
If a function f(x) is discontinuous at x = c, but exists.
Essential discontinuity.
If a function f(x) is discontinuous at x = c and does not exist.

13. Examples.
Give the location of any discontinuity.