AP Calculus

1005 Continuity (2.3)

General Idea:

General Idea: ________________________________________

We already know the continuity of many functions:

Polynomial (Power), Rational, Radical, Exponential, Trigonometric, and Logarithmic functions

DEFN: A function is continuous on an interval if it is continuous at each point in the interval.

DEFN: A function is continuous at a point IFF

a)

b)

c)

Continuity Theorems

Interior Point: A function is continuous at an interior point of its

domain if

ONE-SIDED CONTINUITY

Endpoint: A function is continuous at a left endpoint of i

limx c

y f x c

y f x a

f x f c

ts domain

if lim

or

continuous at a right endpoint if lim .

x a

x b

f x f a

b f x f b

Continuity on a CLOSED INTERVAL.

Theorem: A function is Continuous on a closed interval if it is continuous at every point in the open interval and continuous from one side at the end points.

Example :The graph over the closed interval [-2,4] is given.

Discontinuity

Continuity may be disrupted by:

(a).

c c

(b).

c

(c).

c

Discontinuity: cont.

Method:

(a).

(b).

(c).

Removable or Essential Discontinuities

Examples:

EX:2

( )4

xf x

x

Identify the x-values (if any) at which f(x)is not continuous. Identify the reason for the discontinuity and the type of discontinuity. Is the discontinuity removable or essential?

removable or essential?

Examples: cont.

2

1( )

( 3)f x

x

Identify the x-values (if any) at which f(x)is not continuous. Identify the reason for the discontinuity and the type of discontinuity. Is the discontinuity removable or essential?

removable or essential?

Examples: cont.

3 , 1( )

3 , 1

x xf x

x x

Identify the x-values (if any) at which f(x)is not continuous. Identify the reason for the discontinuity and the type of discontinuity. Is the discontinuity removable or essential?

Graph:

Determine the continuity at each point. Give the reason and the type of discontinuity.

x = -3

x = -2

x = 0

x =1

x = 2

x = 3

Algebraic Method

2

3 2 2( )

3 4 2

x xf x

x x

a.

b.

c.

Algebraic Method 2

2

2

1- 1

( ) - 2 1 3

9 3

3

x x

f x x x

xx

x

At x=1a.

b.

c.

At x=3a.

b.

c.

Consequences of Continuity:

A. INTERMEDIATE VALUE THEOREM

** Existence Theorem

EX: Verify the I.V.T. for f(c) Then find c.

on 2( )f x x 1,2 ( ) 3f c

Consequences: cont.

EX: Show that the function has a ZERO on the interval [0,1].3( ) 2 1f x x x

I.V.T - Zero Locator Corollary

CALCULUS AND THE CALCULATOR:

The calculator looks for a SIGN CHANGE between Left Bound and Right Bound

Consequences: cont.

EX: ( 1)( 2)( 4) 0x x x

I.V.T - Sign on an Interval - Corollary(Number Line Analysis)

EX:1 3

1 2x

Consequences of Continuity:

B. EXTREME VALUE THEOREM

On every closed interval there exists an absolute maximum value and minimum value.

x

y

x

y

Updates:

8/22/10