1-4: Continuity and One- Sided Limits Objectives: Define and explore properties of continuity Discuss one-sided limits Introduce Intermediate Value Theorem.

1-4: Continuity and One-Sided LimitsObjectives:•Define and explore properties of continuity•Discuss one-sided limits•Introduce Intermediate Value Theorem

©2002 Roy L. Gover (roygover@att.net)

Definitionf(x) is continuous at x=c if and only if there are no holes, jumps, skips or gaps in the graph of f(x) at c.

Examples

Continuous Functions

ExamplesDiscontinuous FunctionsRemovable discontinuityJump Discontinuity (non-

removable)Infinite discontinuity (non-removable)

Definition

f(x) is continuous at x=c if and only if:1. f (c) is defined …and

lim ( )x c

f x

2. exists …and

lim ( ) ( )x c

f x f c

3.

Examples

x=2

Discontinuous at x=2 because f(2) is not defined

Examples

x=2

Discontinuous at x=2 because, although f(2) is defined,

2lim ( ) (2)x

f x f

Definition

f(x) is continuous on the open interval (a,b) if and only if f(x) is continuous at every point in the interval.

Try ThisFind the values of x (if any) where f is not continuous. Is the discontinuity removable?

2

0, for 0

, for 0

x

x x

Continuous for all x

( )f x

Try ThisFind the values of x (if any) where f is not continuous. Is the discontinuity removable?1

( )f xx

Discontinuous at x=o, not removable

Definition

f(x) is continuous on the closed interval [a,b] iff it is continuous on (a,b) and continuous from the right at a and continuous from the left at b.

Example

a

b

f(x)

f(x) is continuous on (a,b)

f(x) is continuous from the right at a

f(x) is continuous from the left at b

f(x) is continuous on [a,b]

Definition

lim ( )x c

f x L

is a limit from the right which means x c from

values greater than c

Definition

lim ( )x c

f x L

is a limit from the left which means x c from

values less than c

ExampleFind the limit of f(x) as x approaches 1 from the right:

( ) 1f x x

ExampleFind the limit of f(x) as x approaches 1 from the left:

( ) 1f x x

ExampleFind the limit of f(x) as x approaches 1:

( ) 1f x x

Important IdeaTheorem 1.10:

lim ( )x c

f x

exists iff

lim ( ) lim ( )x c x c

f x f x

Try This

Use the graph to determine the limit, the limit from the right & the limit from the left as x0.

Try This

Use the graph to determine the limit, the limit from the right & the limit from the left as x1.

x=1

Intermediate Value Theorem

Theorem 1.13: If f is continuous on [a,b] and k is a number between f(a) & f(b), then there exists a number c between a & b such that f(c ) =k.

Intermediate Value Theorem

a

f(a)

bf(b)

k

c

Intermediate Value Theorem•an existence theorem; it

guarantees a number exists but doesn’t give a method for finding the number.•it says that a continuous function never takes on 2 values without taking on all the values between.

ExampleRyan was 20 inches long when born and 30 inches long when 9 months old. Since growth is continuous, there was a time between birth and 9 months when he was 25 inches long.

Try ThisUse the Intermediate Value Theorem to show that 3( )f x x

has a zero in the interval [-1,1].

Solution3( )f x x

( 1) 1

(1) 1

f

f

therefore, by the Intermediate Value Theorem, there must be a f (c)=0 where

1 1c

Lesson Close

Tell me one thing you know about continuity and discontinuity.

Assignment

92/1-6 all,7-19 odd,27-49 odd,57,59,65, 81-84