ContinuityContinuity

TS: Making decisions after TS: Making decisions after reflection and reviewreflection and review

ObjectivesObjectives

To find the intervals on which a To find the intervals on which a function is continuous.function is continuous.

To find any discontinuities of a To find any discontinuities of a function.function.

To determine whether discontinuities To determine whether discontinuities are removable or non-removable.are removable or non-removable.

Video Clip fromVideo Clip fromCalculus-Calculus-Help.comHelp.com

ContinuityContinuity

What makes a function What makes a function continuous?continuous?

Continuous functions are predictable…Continuous functions are predictable…

1) No 1) No breaksbreaks in the graph in the graph

A limit must exist at every A limit must exist at every xx-value or -value or the the graph will break.graph will break.

2) No 2) No holesholes or or jumpsjumps

The function cannot have undefined The function cannot have undefined points or vertical asymptotes.points or vertical asymptotes.

ContinuityContinuity

Key PointKey Point: :

Continuous Continuous functions can be functions can be drawn with a drawn with a single, unbroken single, unbroken pencil strokepencil stroke..

ContinuityContinuity

Mathematically speaking…Mathematically speaking…

If If f f ((xx)) is is continuouscontinuous, then for every , then for every x x = c= c in the function, in the function,

In other words, if you can evaluate In other words, if you can evaluate any limit on the function using only any limit on the function using only the the substitutionsubstitution method, then the method, then the function is continuous.function is continuous.

lim ( ) ( )x c

f x f c

Continuity of Polynomial and Continuity of Polynomial and Rational FunctionsRational Functions

A A polynomialpolynomial function is continuous function is continuous at every real number.at every real number.

A A rationalrational function is continuous at function is continuous at every real number in its every real number in its domaindomain. .

Polynomial FunctionsPolynomial Functions3( )f x x x 2( ) 2 3f x x x

Both functions are continuous on . Both functions are continuous on . ( , )

Rational FunctionsRational Functions1

( )f xx

2 1

( )1

xf x

x

continuous on: continuous on: continuous on: continuous on: ( , 0) (0, ) ( , 1) (1, )

Rational FunctionsRational Functions

continuous on: continuous on: continuous on: continuous on: ( , ) ( , 1) ( 1, 1) (1, )

2

1( )

1f x

x

2

1( )

1f x

x

Piecewise FunctionsPiecewise Functions2 4, 2

( )2, 2

x xf x

x x

continuous on continuous on ( , )

22 4 4 4 0

2 2 0

DiscontinuityDiscontinuity

DiscontinuityDiscontinuity: a : a point at which a point at which a function is not function is not continuouscontinuous

DiscontinuityDiscontinuity

Two Types of DiscontinuitiesTwo Types of Discontinuities

1) 1) RemovableRemovable (hole in the graph) (hole in the graph)

2) 2) Non-removableNon-removable (break or vertical (break or vertical asymptote)asymptote)

A discontinuity is A discontinuity is calledcalled removableremovable if if a function can be made continuous a function can be made continuous by defining (or redefining) a point.by defining (or redefining) a point.

Two Types of DiscontinuitiesTwo Types of Discontinuities

DiscontinuityDiscontinuity

2

2( )

3 10

xf x

x x

Find the intervals on which these function Find the intervals on which these function are continuous. are continuous.

2

( 2)( 5)

x

x x

1

( 5)x

Point of discontinuity:

2 0x 2x

Vertical Asymptote:

5 0x 5x

Removablediscontinuity

Non-removablediscontinuity

DiscontinuityDiscontinuity

2

2( )

3 10

xf x

x x

( , 2) ( 2, 5) (5, ) Continuous on:

DiscontinuityDiscontinuity

2

2 , 2( )

4 1, 2

x xf x

x x x

2lim( 2 )x

x

2

2lim( 4 1)x

x x

(2)f

4

3

4( , 2] (2, ) Continuous on:

DiscontinuityDiscontinuity

Determine the value(s) of Determine the value(s) of xx at which the at which the function is discontinuous. Describe the function is discontinuous. Describe the discontinuity as removable or non-discontinuity as removable or non-removable.removable. 2

2

1( )

5 6

xf x

x x

2

2

4 5( )

25

x xf x

x

2

2

10 9( )

81

x xf x

x

2

2

4( )

2 8

xf x

x x

(A) (B)

(C) (D)

DiscontinuityDiscontinuity

2

2

1( )

5 6

xf x

x x

(A)

( 1)( 1)

( 6)( 1)

x x

x x

1x 6x

Removable discontinuity

Non-removable discontinuity

DiscontinuityDiscontinuity

(B)

9x 9x

Removable discontinuity

Non-removable discontinuity

( 9)( 1)

( 9)( 9)

x x

x x

2

2

10 9( )

81

x xf x

x

DiscontinuityDiscontinuity

(C)

5x 5x

Removable discontinuity

Non-removable discontinuity

( 5)( 1)

( 5)( 5)

x x

x x

2

2

4 5( )

25

x xf x

x

DiscontinuityDiscontinuity

(D)

2x 4x

Removable discontinuity

Non-removable discontinuity

( 2)( 2)

( 4)( 2)

x x

x x

2

2

4( )

2 8

xf x

x x

ConclusionConclusion

ContinuousContinuous functions have no breaks, functions have no breaks, no holes, and no jumps.no holes, and no jumps.

If you can evaluate any limit on the If you can evaluate any limit on the function using only the function using only the substitutionsubstitution method, then the function is method, then the function is continuouscontinuous..

ConclusionConclusion

A A discontinuitydiscontinuity is a point at which a is a point at which a function is not continuous.function is not continuous.

Two types of discontinuitiesTwo types of discontinuities RemovableRemovable (hole in the graph) (hole in the graph) Non-removableNon-removable (break or (break or verticalvertical

asymptoteasymptote))