Download - Continuity 1401 x

8/13/2019 Continuity 1401 x

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A functionfis continuous at a pointx = cif the graph offhas no gap, break, split, holes, jumps,

or a missing point forf(x) at the pointx = c. In addition, a functionfis continuous at a pointx = cif you can move along the graph of fthrough the point (c, f(c)) with a pencil without lifting

the pencil from the paper.

If a functionfis continuous at a pointx = c, then thecx

limf(x) must exists and its value must

eual f(c).

!or a function to be continuous at the pointx = c, the function must have a value atx = c(which

is f(c)) and thecx

limf(x) must have a value which is thesame asthe value of the function atx =

c. In general, a function is said to be continuous on the interval "a, b# if it is continuous at each

point in the interval.

It is important to understand what it means to say thatcx

limf(x) exists.

$%&&' ichael Aryee continuity age *

The concept of continuity

A function fis continuous at a pointx = cif all of the following conditions are fulfilled+

*. f(c ) is defined (this is means that the value of f(c) is not missing or undefined).

%.cx

limf(x) exists, (this is means that the value ofcx

limf(x) is not missing or

undefined),

and

.cx

limf(x) -f(c) (this is means that the limit,cx

lim f(x), exists and its value is eual

to f(c))

cx

limf(x) exists if the following all of the following statements are true

*. cx

limf(x) is definite and has a value.

%. +cx

lim f(x) is definite and has a value.

. cx

limf(x) - +cx

lim f(x).


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When a function is not defined at certain values, these values are certainly the points of

discontinuity of the function. /ecause division by 0ero is not allowed in mathematics, anynumber or numbers that when substituted into the function causes the denominator to be 0ero,

thus making the function undefined, represent points of discontinuity.

1e usually concentrate only on either the domain of the function or the endpoints of the interval

making the domain.

2here are two main categories of discontinuity+ removableand non-removable. 3on4removable

usually occurs where the graph has a break in it. 5ome people refer to this kind of discontinuityasjump discontinuity.

$%&&' ichael Aryee continuity age %

Points of discontinuity

Eample of discontinuity

A functionfis discontinuousat a pointx = cif one of the following conditions are fulfilled+

*. f(c ) is not defined at c,

%.cx

limf(x) does not exist, or

.cx

limf(x) f(c), (this is the case where the limit,cx

limf(x), exists but not eual to f(c))

A function is said to be discontinuousatx = c, if a hole, gap, or break occurs in the graph at

x = c, meaning the function violates one of the three items above.

o A non-removable discontinuityis the situation where we are unableto find a value

forcx

limf(x) (that is,cx

limf(x) is not defined at c).

o A removable discontinuityis the situation where

*. 1e are able to find a value forcx

limf(x), in this casecx

limf(x) exists, but we

are unableto find a value for f(c), or

%. 1e are able to get values for bothcx

limf(x) and f(c), however, their values do

not match, that is,cx

limf(x) f(c).

The criterion for deciding whether a function is removable or irremovable is based on

whether the limit of the function at the point in question exists or does not exist.


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6iven the graph off(x)shown below determine iff(x)is continuous at a)x=& and b) x=*.

a) atx=&

!eft-hand !imit "i#ht-hand !imit T$o-sided !imit %alue of &unction

&limx

f(x) - & +&

limx

f(x) - &

!rom 7eft4hand and 8ight

hand we conclude that+

&limx

f(x) - &

Atx=&,

f(&) - *

'onclusion:5ince the limit at the point and the function value at that point are not eual, the

function isn9t continuous at that point. 2he function is discontinuous at x - &. 2hat is,&

limx

f(x)

f(&). 2his type of discontinuity is called removable since the limit exists. It is called

removable discontinuity because we can make the discontinuity to disappear by changing thevalue offat the point in uestion. 2hat is, we can change the value of f(c) to resolve the problem.

:n a graph, a removable discontinuity can be identified by a hole punched in the graph.

b) atx=*


*limx

f(x) - * +

*limx

f(x) - &



*limx

f(x) - (oes Not

Eist

Atx=*,

f(*) - *


function isn9t continuous at that point. 2he function is discontinuous at x - *. 2hat is,&

limx

f(x)

f(&). 2his type of discontinuity is non-removablesince the limit does not exist.

$%&&' ichael Aryee continuity age

Another eample of discontinuity


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;etermine whether the given function is continuous.

f(x) -

+ ,%

,

x

x

*

*

>

x

x

2he domain of this function is all real numbers. 2his is just one function, but it has differentdescription in different parts of the domain.

:ur task is to look for points of discontinuity. 2hese points may represent a value for which thedefinition of the function changes (in this particular case at x - *) or values that causes the

denominator to be 0ero).

1e call such values suspicious points. Always check for continuity at these suspicious points.

In this problem x - * is a suspicious point, and we must check for continuity at this point.


*

limx f(x) -

*

limx

x

-

+

*

limx f(x) -

+

*

limx

%x

- '



&limx

f(x) - (oes Not

Eist

Atx=*,

f(*) - (*) -


function isn9t continuous at that point. 2he function is discontinuous at x - *.

2hat is,&

limx

f(x) f(*). 2his type of discontinuity is non4removable since the limit does not

exist.

$%&&' ichael Aryee continuity age


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;etermine whether the given function is continuous.

f(x) -

+ ,*%

,

x

x

*

*

>

x

x

2he domain of this function is all real numbers. 2his is just one function, but it has differentdescription in different parts of the domain.

:ur task is to look for points of discontinuity. 2hese points may represent a value for which the

definition of the function changes (in this particular case at x - *) or values that causes thedenominator to be 0ero).

1e call such values suspicious points. Always check for continuity at these suspicious points.

In this problem x - * is a suspicious point, and we must check for continuity at this point.


*limx

f(x) -

*limx

x

-

+*limx

f(x) -+

*limx

%x*

-

!rom 7eft4hand and 8ighthand we conclude that+

*limx

f(x) -

Atx=*,

f(*) - (*) -

'onclusion: 5ince the limit at the point and the function value at that point are eual, the

function is continuous at that point. 2he function is continuous at x - *.

2hat is,*

limx

f(x) -f(*).

$%&&' ichael Aryee continuity age '

An eample of a continuous function


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5tate whether the function is continuous at all points. 6ive the points of discontinuity, if any.

7et f(x) -

=

+

% x,=

%x,%

*&%

x

xx

In this problem x - % is a suspicious point, and we must check for continuity at this point.

2he domain of this function is all real numbers.

!eft-hand !imit "i#ht-hand !imit T$o-sided !imit %alue of

&unction

%limx

f(x) -

%limx

%

*&,%

+

x

xx

-

%limx

)%(

)')(%(

+

x

xx

-

%limx

(x > ') - =

5ame as 7eft4hand

limit+

+%

limx

f(x) - =

!rom 7eft4hand and

8ight hand weconclude that+

%limx

f(x) - =

Atx=%,

f(%) - =

'onclusion:5ince the limit at the point x - % and the function value at that point are eual, thefunction is continuous at that point. 2he function is continuous at x - %.

2hat is,%

limx

f(x) -f(%).

$%&&' ichael Aryee continuity age ?

An eample of a continuous function


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1here is the function f(x) - @ x % @ continuousB

)olution:

2he given function is the same as f(x) -

),%(,%

x

x

%

%