2.7: Continuity and the Intermediate Value Theorem

Objectives:•Define and explore properties of continuity•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

ExampleFind the values of k, if possible, that will make the function continuous.

2 , for 2

2 , for 2

kx x

x k x

( )f x

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]

Graphing calculators can make non-continuous functions appear continuous.

Graph: floory x

CATALOG F floor(

Note resolution.

The calculator “connects the dots” which covers up the discontinuities.

Graphing calculators can make non-continuous functions appear continuous.

Graph: floory x

CATALOG F floor(

GRAPH

The open and closed circles do not show, but we can see the discontinuities.

If we change the plot style to “dot” and the resolution to 1, then we get a graph that is closer to the correct floor graph.

Intermediate Value Theorem

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.

ExampleKaley was 20 inches long when born. Let’s say that she will be 30 inches long when 15 months old. Since growth is continuous, there was a time between birth and 15 months when she 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