1. Continuity
In section 2.3 we noticed that the limit of a function as x approaches a can often be found simply by calculating the value of the function at a. Please keep in mind throughout this lesson that “a continuous process is one that takes place gradually, without interruption or abrupt change”.

2. Continuity
Definition: A function f is continuous at a number a if If f is not continuous at a, we say f is discontinuous at a, or f has a discontinuity at a.

3. Continuity
This definition requires three things if f is continuous at a: f(a) is defined (that is, a is in the domain of f) 3.

4. Continuity
Looking at the graph of f from the ELMO,  At which numbers is f discontinuous? Why?

5. Examples of continuous functions in real life:
Continuity
Examples of continuous functions in real life:
The displacement or velocity of a vehicle varies continuously with time.
A person’s height varies continuously with time.
Geometrically: A graph that has no break in it. The graph can be drawn without removing your pen from the paper.

6. Continuity
Now let’s see how to detect discontinuities when a function is defined by a formula…

7. Continuity
Where are each of the following functions discontinuous? f(x) =

8. Continuity
4. 5.

9. Continuity
Notice the graphs on page None of these graphs can be traced without lifting your pencil.

10. Continuity
Definition: A function f is continuous from the right at a number a if and f is continuous from the left at a if Look at example 3 on page 121…

11. Continuity
Definition: A function f is continuous on an interval if it is continuous at every number in the interval. (If f is defined only on one side of an endpoint of the interval, we understand continuous at the endpoint to mean continuous from the right or continuous from the left)

12. Continuity
Example: Show that the function is continuous on the interval [-1, 1].

13. Continuity
Theorem: If f and g are continuous at a and c is a constant, then the following functions are also continuous at a:
f + g f – g
cf fg 5.

14. Continuity
Theorem: a) Any polynomial is continuous everywhere; that is, it is continuous on all real numbers (-∞, ∞). b) Any rational function is continuous wherever it is defined; that is, it is continuous on its domain. It turns out that most of the familiar functions are continuous at every number in their domains.

15. Continuity
Now lets take a moment to examine the sine and cosine… Recalling the appearance of the sine and cosine graphs, we say that these functions are continuous. We know that the coordinates of the point P on the unit circle is (cosø, sinø). As ø 0, we can demonstrate that P approaches the point (1, 0). Therefore: and

16. Continuity
Theorem: The following types of functions are continuous at every number in their domains: polynomials rational functions root functions trigonometric functions inverse trigonometric functions exponential functions logarithmic functions

17. Continuity
Where is the function continuous?

18. Continuity
Example: Evaluate

19. Continuity
Theorem: If f is continuous at b and , then In other words,

20. Continuity
Evaluate

21. Continuity
Theorem: If g is continuous at a and f is continuous at g(a), then the composite function f g given by (f g)(x) = f(g(x)) is continuous at a. “A continuous function of a continuous function is a continuous function.”

22. Continuity Example: Where are the following functions continuous?
h(x) = sin(x2)
F(x) = ln(1 + cosx)

23. Continuity
THE INTERMEDIATE VALUE THEOREM: Suppose that f is continuous on the closed interval [a, b] and let N be any number between f(a) and f(b). Then there exists a number c in (a, b) such that f(c) = N. “The Intermediate Value Theorem states that a continuous function takes on every intermediate value between the function values f(a) and f(b). LOOK AT FIGURE 8 ON PAGE 127.

24. Continuity
One use of the Intermediate Value Theorem is in locating roots of equations… Example: Show that there is a root of the equation 4x3 – 6x2 + 3x – 2 = 0 between 1 and 2.

25. Continuity
ASSIGNMENT: P. 128 (1-6, 10, 13-21, 29, 35-38)