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Continuity

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Continuity of a Function at a NumberA function f is continuous at a if

1. f is defined at x = a ;that is, a is in the domain of f so the value of f (a ) exists.

)(lim x f a x

2. exists;

that is,

3. )()(lim a f x f a x

)()( limlim x f x f a xa x

If at least one of the three conditions fail tohold at a , then the function is said to be

discontinuous at a .

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c

y= f ( x)

y

x

f (c)

Continuous

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c

y= f ( x)

y

x

f (c)

Not continuous

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c

y= f ( x)

y

x

Not continuous

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c

y= f ( x)

y

x

Not continuous

f (c)

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c

y= f ( x)

y

x

Not continuous

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c

y= f ( x)

y

x

Not continuous

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Continuity of Rational and PolynomialFunctions

A polynomial function is continuous everywhere.

A rational function defined by

where N( x) and D( x) are both polynomial functions, iscontinuous at every number a provided that .0)( x D

)(

)()(

x D x N

x f

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Consider the function defined by

Example:

Determine if f is continuousat x = 3.

3,3

3,)(

2

xif x

xif x x x f

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Consider the function defined by

Example:

Determine if f is continuous at x = 2 andat x = 1 .

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Continuity on an IntervalA function f is said to be continuous

)()(lim a f x f a x

on an open interval if and only if it is continuous atevery number in the open interval.

on the half-open interval [ a, b) if and only if it is

continuous on the open interval ( a , b) and

on the half-open interval (a, b] if and only if it iscontinuous on the open interval ( a , b) and

)()(lim b f x f b x

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Continuity on an Interval

A function f is said to be continuous

)()(lim a f x f a x

on the closed interval [a , b] if and only if it iscontinuous on the open interval ( a , b) and

)()(lim b f x f b x

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For each of the given functions, do thefollowing:

A. Sketch the graph of the givenfunction.

B. By observing where there is a breakin the graph, determine the numberat which the function isdiscontinuous.

C. Verify, using the definition of continuity, the 3 conditions to

determine the continuity of thefunction at the possible point(s) of discontinuity.

D. Find the largest interval or union of intervals for which the given functionis continuous.

Exercises:

1,3

1,3)( x x

x x x f

3,2

3,3)(

x

x x xg

2)(

x x

xh

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For each of the given functions, do thefollowing:

A. Sketch the graph of the givenfunction.

B. By observing where there is a breakin the graph, determine the numberat which the function isdiscontinuous.

C. Verify, using the definition of continuity, the 3 conditions to

determine the continuity of thefunction at the possible point(s) of discontinuity.

D. Find the largest interval or union of intervals for which the given functionis continuous.

Exercises:

25

21,321,

)(

2

x x

x x x x

xg

14

11,41,32

)( 2

x x

x x x x x

x f

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For each of the given functions, do thefollowing:

A. Sketch the graph of the givenfunction.

B. By observing where there is a breakin the graph, determine the numberat which the function isdiscontinuous.

C. Verify, using the definition of continuity, the 3 conditions to

determine the continuity of thefunction at the possible point(s) of discontinuity.

D. Find the largest interval or union of intervals for which the given functionis continuous.

Exercises:

228

28

2,23

)(2 x x x

x

x x

xg

2,1

2,4)(2

2

x x x

x x x x f