Practice!

1. For the graph shown, which of these statements is FALSE?

(A) f(x) is continuous at x=2(B) (C)

(D)

(E) f(x) is continuous everywhere from

2.

 

1lim ( ) 2x

f x

0lim ( ) 3x

f x

2 4x

3 2

3 2

2 3 4lim

4 3 2 1x

x x x

x x x

1

( ) 4 ( ) 1 ( ) ( )0 ( ) 14

A B C D E

AP Calculus Unit 1 Day 7

Continuity, Composition, & Intermediate Value Theorem

Three Types of Discontinuity—Be Familiar with the Vocabulary

1. Hole when factors divide out “Removable discontinuity”

2. Asymptotes when denominator = 0 “Infinite discontinuity, nonremovable”

3. Piecewise Graph or Greatest Integer Function “Jump Discontinuity, also nonremovable”

f(x) is continuous at x=c if . . .

• Official Definition

• Which means . . .

limx c

f x f c

Is f(x) continuous at x = 4?Justify your answer

3 4

5 4

x xf x

x x

Is f(x) continuous at x = 3?Justify your answer

2 3

3 3

x xf x

x x

“Remove” the discontinuity in this function by rewriting the function as a piecewise function.

2 2 8( )

2

x xf x

x

Intermediate Value Thm.

A continuous function takes on all y values between f(a) and f(b).

As Related to The Preview Video

If W is between f(a) & f(b), then W = f(c) for some c in [a,b]

Graphical Illustration of the Intermediate Value Theorem

a b

f(b)

f(a)

Any W value in here will be “hit” at least once

Intermediate Value Theorem in Action

Given f(x) is continuous on the interval [0,4].

What is the minimum number of times f(x) = 8? Justify.x 0 1 2 3 4

f(x) 10 6 2 7 9

Composition of functions

Composition of functions. . .Is f(g(x)) continuous at x = 4?

We must evaluate

Therefore f(g(x)) is continuous

Find the limit of the composition

2

2

( ) 3 2 ( )

Find lim ( ( ))x

f x x g x x

f g x

Find the limit of the composition

2

2

( 2)( ) 3 2 ( )

2

Find lim ( ( ))x

x xf x x g x

x

f g x

Find the limit of the composition

2

2

3

4 2 3 1( ) ( )

5 3

Find lim ( ( ))x

x xf x g x

x x

f g x