1. Continuity

2. What you’ll learn about
Continuity at a Point
Continuous Functions
Algebraic Combinations
Composites
Intermediate Value Theorem for Continuous Functions
Essential Question
How can continuous functions be used to describe how a body moves through space and how the speed of a chemical reaction changes with time?

3. Continuity at a Point Example Continuity at a Point
Find the points at which the given function is continuous and the points at which it is discontinuous.
The function is continuous at 0 < x < 2, and 3 < x < 6.
The function is discontinuous at x < 0, 2 < x < 3, and x > 6.

4. Continuity at a Point
If a function f is not continuous at a point c , we say that f is discontinuous at c and c is a point of discontinuity of f.
Note that c need not be in the domain of f.

5. Continuity at a Point Removable Jump Infinite Oscillating
The typical discontinuity types are:
Removable
Jump
Infinite
Oscillating

6. Example Continuity at a Point
Given the graph of f (x), shown below, determine if f (x) is continuous at x = -2, x = 0, x = 3. Then name the type of discontinuity at each point.
Discontinuous at x = – 2
Continuous at x = 0
Jump Disc.
Discontinuous at x = 3
Removable Disc.

7. Example Continuity at a Point
Write a piecewise function for the given function to remove any point of removable discontinuity.

8. Continuous Functions
A function is continuous on an interval if and only if it is continuous at every point of the interval. A continuous function is one that is continuous at every point of its domain. A continuous function need not be continuous on every interval.
Is the following function a continuous function?
The function is a continuous function because it is continuous at every point of its domain.
It does have a point of discontinuity at x = – 1 because it is not defined there.

9. Properties of Continuous Functions

10. Composite of Continuous Functions
Use the Theorem to show that the given function is continuous?
Since we know the sin function is continuous and the polynomial f (x) is continuous, the composition function is continuous.

11. Intermediate Value Theorem
Suppose that f (x ) is continuous on [a, b] and let M be any number between f (a ) and f (b ).   Then there exists a number c such that,
The Intermediate Value Theorem tells us that a function will take the value of M somewhere between a and b but it doesn’t tell us where it will take the value nor does it tell us how many times it will take the value.  These are important ideas to remember about the Intermediate Value Theorem.

12. Intermediate Value Theorem
Show that the following function has a root somewhere in the interval [-1, 2].
Since we know polynomial p (x) is continuous, by the Intermediate Value Theorem, there must be values of x in the given interval that have outputs between –19 and 8. Therefore there is a value of x in the interval that will have an output of 0.

13. Intermediate Value Theorem
If possible, determine if the following function takes the following values in the interval [0, 5].
(a) Does f (x) = 10?
(b) Does f (x) = –10?
yes
Cannot determine