Lesson 5: Continuity
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Transcript of Lesson 5: Continuity
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Section1.5Continuity
V63.0121.027, CalculusI
September17, 2009
Announcements
I PleaseputyourhomeworkintheenvelopebylastnameI QuiznextweekinRecitationon§§1.1–1.3I WebAssignments1and2dueTuesday
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. . . . . .
Hatsumon
Herearesomediscussionquestionstostart.I Wereyoueverexactlythreefeettall?
I Wasyourheight(ininches)everequaltoyourweight(inpounds)?
I Isthereapairofpointsonoppositesidesoftheworldatthesametemperatureatthesametime?
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. . . . . .
Hatsumon
Herearesomediscussionquestionstostart.I Wereyoueverexactlythreefeettall?I Wasyourheight(ininches)everequaltoyourweight(inpounds)?
I Isthereapairofpointsonoppositesidesoftheworldatthesametemperatureatthesametime?
![Page 4: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/4.jpg)
. . . . . .
Hatsumon
Herearesomediscussionquestionstostart.I Wereyoueverexactlythreefeettall?I Wasyourheight(ininches)everequaltoyourweight(inpounds)?
I Isthereapairofpointsonoppositesidesoftheworldatthesametemperatureatthesametime?
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. . . . . .
Outline
Continuity
TheIntermediateValueTheorem
BacktotheQuestions
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. . . . . .
Recall: DirectSubstitutionProperty
Theorem(TheDirectSubstitutionProperty)If f isapolynomialorarationalfunctionand a isinthedomainoff, then
limx→a
f(x) = f(a)
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. . . . . .
DefinitionofContinuity
Definition
I Let f beafunctiondefinednear a. Wesaythat f is continuousat aif
limx→a
f(x) = f(a).
I A function f iscontinuous ifitiscontinuousateverypointinitsdomain.
. .x
.y
.
.a
.f(a)
Therearethreeimportantpartstothisdefinition.I ThelimithastoexistI thefunctionhastobedefinedI andthesevalueshavetoagree.
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. . . . . .
DefinitionofContinuity
Definition
I Let f beafunctiondefinednear a. Wesaythat f is continuousat aif
limx→a
f(x) = f(a).
I A function f iscontinuous ifitiscontinuousateverypointinitsdomain.
. .x
.y
.
.a
.f(a)
Therearethreeimportantpartstothisdefinition.I ThelimithastoexistI thefunctionhastobedefinedI andthesevalueshavetoagree.
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. . . . . .
FreeTheorems
Theorem
(a) Anypolynomialiscontinuouseverywhere; thatis, itiscontinuouson R = (−∞,∞).
(b) Anyrationalfunctioniscontinuouswhereveritisdefined;thatis, itiscontinuousonitsdomain.
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. . . . . .
Showingafunctioniscontinuous
ExampleLet f(x) =
√4x + 1. Showthat f iscontinuousat 2.
SolutionWewanttoshowthat lim
x→2f(x) = f(2). Wehave
limx→a
f(x) = limx→2
√4x + 1 =
√limx→2
(4x + 1) =√9 = 3 = f(2).
Eachstepcomesfromthelimitlaws.
QuestionAtwhichotherpointsis f continuous?
AnswerThefunction f iscontinuouson (−1/4,∞). Itis rightcontinuousatthepoint −1/4 since lim
x→−1/4+f(x) = f(−1/4).
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. . . . . .
Showingafunctioniscontinuous
ExampleLet f(x) =
√4x + 1. Showthat f iscontinuousat 2.
SolutionWewanttoshowthat lim
x→2f(x) = f(2). Wehave
limx→a
f(x) = limx→2
√4x + 1 =
√limx→2
(4x + 1) =√9 = 3 = f(2).
Eachstepcomesfromthelimitlaws.
QuestionAtwhichotherpointsis f continuous?
AnswerThefunction f iscontinuouson (−1/4,∞). Itis rightcontinuousatthepoint −1/4 since lim
x→−1/4+f(x) = f(−1/4).
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. . . . . .
Showingafunctioniscontinuous
ExampleLet f(x) =
√4x + 1. Showthat f iscontinuousat 2.
SolutionWewanttoshowthat lim
x→2f(x) = f(2). Wehave
limx→a
f(x) = limx→2
√4x + 1 =
√limx→2
(4x + 1) =√9 = 3 = f(2).
Eachstepcomesfromthelimitlaws.
QuestionAtwhichotherpointsis f continuous?
AnswerThefunction f iscontinuouson (−1/4,∞). Itis rightcontinuousatthepoint −1/4 since lim
x→−1/4+f(x) = f(−1/4).
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. . . . . .
Showingafunctioniscontinuous
ExampleLet f(x) =
√4x + 1. Showthat f iscontinuousat 2.
SolutionWewanttoshowthat lim
x→2f(x) = f(2). Wehave
limx→a
f(x) = limx→2
√4x + 1 =
√limx→2
(4x + 1) =√9 = 3 = f(2).
Eachstepcomesfromthelimitlaws.
QuestionAtwhichotherpointsis f continuous?
AnswerThefunction f iscontinuouson (−1/4,∞).
Itis rightcontinuousatthepoint −1/4 since lim
x→−1/4+f(x) = f(−1/4).
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. . . . . .
Showingafunctioniscontinuous
ExampleLet f(x) =
√4x + 1. Showthat f iscontinuousat 2.
SolutionWewanttoshowthat lim
x→2f(x) = f(2). Wehave
limx→a
f(x) = limx→2
√4x + 1 =
√limx→2
(4x + 1) =√9 = 3 = f(2).
Eachstepcomesfromthelimitlaws.
QuestionAtwhichotherpointsis f continuous?
AnswerThefunction f iscontinuouson (−1/4,∞). Itis rightcontinuousatthepoint −1/4 since lim
x→−1/4+f(x) = f(−1/4).
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. . . . . .
TheLimitLawsgiveContinuityLaws
TheoremIf f and g arecontinuousat a and c isaconstant, thenthefollowingfunctionsarealsocontinuousat a:
I f + gI f− gI cfI fg
I fg(if g(a) ̸= 0)
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. . . . . .
Whyasumofcontinuousfunctionsiscontinuous
Wewanttoshowthat
limx→a
(f + g)(x) = (f + g)(a).
Wejustfollowournose:
limx→a
(f + g)(x) = limx→a
[f(x) + g(x)]
= limx→a
f(x) + limx→a
g(x) (iftheselimitsexist)
= f(a) + g(a) (theydo; f and g arects)
= (f + g)(a)
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. . . . . .
Trigonometricfunctionsarecontinuous
I sin and cos arecontinuouson R.
I tan =sincos
and sec =1cos
arecontinuousontheirdomain, whichisR \
{ π
2+ kπ
∣∣∣ k ∈ Z}.
I cot =cossin
and csc =1sin
arecontinuousontheirdomain, whichisR \ { kπ | k ∈ Z }.
..sin
.cos
.tan .sec
.cot .csc
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. . . . . .
Trigonometricfunctionsarecontinuous
I sin and cos arecontinuouson R.
I tan =sincos
and sec =1cos
arecontinuousontheirdomain, whichisR \
{ π
2+ kπ
∣∣∣ k ∈ Z}.
I cot =cossin
and csc =1sin
arecontinuousontheirdomain, whichisR \ { kπ | k ∈ Z }.
..sin
.cos
.tan .sec
.cot .csc
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. . . . . .
Trigonometricfunctionsarecontinuous
I sin and cos arecontinuouson R.
I tan =sincos
and sec =1cos
arecontinuousontheirdomain, whichisR \
{ π
2+ kπ
∣∣∣ k ∈ Z}.
I cot =cossin
and csc =1sin
arecontinuousontheirdomain, whichisR \ { kπ | k ∈ Z }.
..sin
.cos
.tan
.sec
.cot .csc
![Page 20: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/20.jpg)
. . . . . .
Trigonometricfunctionsarecontinuous
I sin and cos arecontinuouson R.
I tan =sincos
and sec =1cos
arecontinuousontheirdomain, whichisR \
{ π
2+ kπ
∣∣∣ k ∈ Z}.
I cot =cossin
and csc =1sin
arecontinuousontheirdomain, whichisR \ { kπ | k ∈ Z }.
..sin
.cos
.tan .sec
.cot .csc
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. . . . . .
Trigonometricfunctionsarecontinuous
I sin and cos arecontinuouson R.
I tan =sincos
and sec =1cos
arecontinuousontheirdomain, whichisR \
{ π
2+ kπ
∣∣∣ k ∈ Z}.
I cot =cossin
and csc =1sin
arecontinuousontheirdomain, whichisR \ { kπ | k ∈ Z }.
..sin
.cos
.tan .sec
.cot
.csc
![Page 22: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/22.jpg)
. . . . . .
Trigonometricfunctionsarecontinuous
I sin and cos arecontinuouson R.
I tan =sincos
and sec =1cos
arecontinuousontheirdomain, whichisR \
{ π
2+ kπ
∣∣∣ k ∈ Z}.
I cot =cossin
and csc =1sin
arecontinuousontheirdomain, whichisR \ { kπ | k ∈ Z }.
..sin
.cos
.tan .sec
.cot .csc
![Page 23: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/23.jpg)
. . . . . .
ExponentialandLogarithmicfunctionsarecontinuous
Foranybase a > 1,
I thefunction x 7→ ax iscontinuouson R
I thefunction loga iscontinuousonitsdomain: (0,∞)
I Inparticular ex andln = loge arecontinuousontheirdomains
.
.ax
.loga x
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. . . . . .
ExponentialandLogarithmicfunctionsarecontinuous
Foranybase a > 1,
I thefunction x 7→ ax iscontinuouson R
I thefunction loga iscontinuousonitsdomain: (0,∞)
I Inparticular ex andln = loge arecontinuousontheirdomains
.
.ax
.loga x
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. . . . . .
ExponentialandLogarithmicfunctionsarecontinuous
Foranybase a > 1,
I thefunction x 7→ ax iscontinuouson R
I thefunction loga iscontinuousonitsdomain: (0,∞)
I Inparticular ex andln = loge arecontinuousontheirdomains
.
.ax
.loga x
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. . . . . .
InversetrigonometricfunctionsaremostlycontinuousI sin−1 and cos−1 arecontinuouson (−1,1), leftcontinuousat 1, andrightcontinuousat −1.
I sec−1 and csc−1 arecontinuouson (−∞,−1) ∪ (1,∞), leftcontinuousat −1, andrightcontinuousat 1.
I tan−1 and cot−1 arecontinuouson R.
.
.−π
.−π/2
.π/2
.π
.
.
.sin−1
.
.
.cos−1.sec−1
.
..csc−1
.
.
.tan−1
.cot−1
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. . . . . .
InversetrigonometricfunctionsaremostlycontinuousI sin−1 and cos−1 arecontinuouson (−1,1), leftcontinuousat 1, andrightcontinuousat −1.
I sec−1 and csc−1 arecontinuouson (−∞,−1) ∪ (1,∞), leftcontinuousat −1, andrightcontinuousat 1.
I tan−1 and cot−1 arecontinuouson R.
.
.−π
.−π/2
.π/2
.π
.
.
.sin−1.
.
.cos−1
.sec−1
.
..csc−1
.
.
.tan−1
.cot−1
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. . . . . .
InversetrigonometricfunctionsaremostlycontinuousI sin−1 and cos−1 arecontinuouson (−1,1), leftcontinuousat 1, andrightcontinuousat −1.
I sec−1 and csc−1 arecontinuouson (−∞,−1) ∪ (1,∞), leftcontinuousat −1, andrightcontinuousat 1.
I tan−1 and cot−1 arecontinuouson R.
.
.−π
.−π/2
.π/2
.π
.
.
.sin−1.
.
.cos−1.sec−1
.
.
.csc−1
.
.
.tan−1
.cot−1
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. . . . . .
InversetrigonometricfunctionsaremostlycontinuousI sin−1 and cos−1 arecontinuouson (−1,1), leftcontinuousat 1, andrightcontinuousat −1.
I sec−1 and csc−1 arecontinuouson (−∞,−1) ∪ (1,∞), leftcontinuousat −1, andrightcontinuousat 1.
I tan−1 and cot−1 arecontinuouson R.
.
.−π
.−π/2
.π/2
.π
.
.
.sin−1.
.
.cos−1.sec−1
.
..csc−1
.
.
.tan−1
.cot−1
![Page 30: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/30.jpg)
. . . . . .
InversetrigonometricfunctionsaremostlycontinuousI sin−1 and cos−1 arecontinuouson (−1,1), leftcontinuousat 1, andrightcontinuousat −1.
I sec−1 and csc−1 arecontinuouson (−∞,−1) ∪ (1,∞), leftcontinuousat −1, andrightcontinuousat 1.
I tan−1 and cot−1 arecontinuouson R.
.
.−π
.−π/2
.π/2
.π
.
.
.sin−1.
.
.cos−1.sec−1
.
..csc−1
.
.
.tan−1
.cot−1
![Page 31: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/31.jpg)
. . . . . .
InversetrigonometricfunctionsaremostlycontinuousI sin−1 and cos−1 arecontinuouson (−1,1), leftcontinuousat 1, andrightcontinuousat −1.
I sec−1 and csc−1 arecontinuouson (−∞,−1) ∪ (1,∞), leftcontinuousat −1, andrightcontinuousat 1.
I tan−1 and cot−1 arecontinuouson R.
.
.−π
.−π/2
.π/2
.π
.
.
.sin−1.
.
.cos−1.sec−1
.
..csc−1
.
.
.tan−1
.cot−1
![Page 32: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/32.jpg)
. . . . . .
Whatcouldgowrong?
Inwhatwayscouldafunction f failtobecontinuousatapoint a?Lookagainatthedefinition:
limx→a
f(x) = f(a)
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. . . . . .
Pitfall#1
: Thelimitdoesnotexist
ExampleLet
f(x) =
{x2 if 0 ≤ x ≤ 1
2x if 1 < x ≤ 2
Atwhichpointsis f continuous?
SolutionAtanypoint a in [0, 2] besides 1, lim
x→af(x) = f(a) because f is
representedbyapolynomialnear a, andpolynomialshavethedirectsubstitutionproperty. However,
limx→1−
f(x) = limx→1−
x2 = 12 = 1
limx→1+
f(x) = limx→1+
2x = 2(1) = 2
So f hasnolimitat 1. Therefore f isnotcontinuousat 1.
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. . . . . .
Pitfall#1: Thelimitdoesnotexist
ExampleLet
f(x) =
{x2 if 0 ≤ x ≤ 1
2x if 1 < x ≤ 2
Atwhichpointsis f continuous?
SolutionAtanypoint a in [0, 2] besides 1, lim
x→af(x) = f(a) because f is
representedbyapolynomialnear a, andpolynomialshavethedirectsubstitutionproperty. However,
limx→1−
f(x) = limx→1−
x2 = 12 = 1
limx→1+
f(x) = limx→1+
2x = 2(1) = 2
So f hasnolimitat 1. Therefore f isnotcontinuousat 1.
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. . . . . .
GraphicalIllustrationofPitfall#1
. .x
.y
..−1
..1
..2
..−1
..1
..2
..3
..4
.
.
.
.
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. . . . . .
Pitfall#2
: Thefunctionhasnovalue
ExampleLet
f(x) =x2 + 2x + 1
x + 1
Atwhichpointsis f continuous?
SolutionBecause f isrational, itiscontinuousonitswholedomain. Notethat −1 isnotinthedomainof f, so f isnotcontinuousthere.
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. . . . . .
Pitfall#2: Thefunctionhasnovalue
ExampleLet
f(x) =x2 + 2x + 1
x + 1
Atwhichpointsis f continuous?
SolutionBecause f isrational, itiscontinuousonitswholedomain. Notethat −1 isnotinthedomainof f, so f isnotcontinuousthere.
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. . . . . .
GraphicalIllustrationofPitfall#2
. .x
.y
...−1
. .1
f cannotbecontinuouswhereithasnovalue.
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. . . . . .
Pitfall#3
: functionvalue ̸= limit
ExampleLet
f(x) =
{7 if x ̸= 1
π if x = 1
Atwhichpointsis f continuous?
Solutionf isnotcontinuousat 1 because f(1) = π but lim
x→1f(x) = 7.
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. . . . . .
Pitfall#3: functionvalue ̸= limit
ExampleLet
f(x) =
{7 if x ̸= 1
π if x = 1
Atwhichpointsis f continuous?
Solutionf isnotcontinuousat 1 because f(1) = π but lim
x→1f(x) = 7.
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. . . . . .
GraphicalIllustrationofPitfall#3
. .x
.y
..π
..7
..1
.
.
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. . . . . .
Specialtypesofdiscontinuites
removablediscontinuity Thelimit limx→a
f(x) exists, but f isnot
definedat a oritsvalueat a isnotequaltothelimitat a.
jumpdiscontinuity Thelimits limx→a−
f(x) and limx→a+
f(x) exist, but
aredifferent. f(a) isoneoftheselimits.
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. . . . . .
Graphicalrepresentationsofdiscontinuities
. .x
.y
..π
..7
..1
.
.
removable
. .x
.y
..−1
..1
..2
..−1
..1
..2
..3
..4
.
.
.
.
jump
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. . . . . .
Thegreatestintegerfunction[[x]] isthegreatestinteger ≤ x.
. .x
.y
..−2
..−2
..−1
..−1
..1
..1
..2
..2
..3
..3
. .
. .
. .
. .
. ..y = [[x]]
Thisfunctionhasajumpdiscontinuityateachinteger.
![Page 45: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/45.jpg)
. . . . . .
Thegreatestintegerfunction[[x]] isthegreatestinteger ≤ x.
. .x
.y
..−2
..−2
..−1
..−1
..1
..1
..2
..2
..3
..3
. .
. .
. .
. .
. ..y = [[x]]
Thisfunctionhasajumpdiscontinuityateachinteger.
![Page 46: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/46.jpg)
. . . . . .
Outline
Continuity
TheIntermediateValueTheorem
BacktotheQuestions
![Page 47: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/47.jpg)
. . . . . .
A BigTimeTheorem
Theorem(TheIntermediateValueTheorem)Supposethat f iscontinuousontheclosedinterval [a,b] andlet Nbeanynumberbetween f(a) and f(b), where f(a) ̸= f(b). Thenthereexistsanumber c in (a,b) suchthat f(c) = N.
![Page 48: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/48.jpg)
. . . . . .
IllustratingtheIVT
Supposethat f iscontinuousontheclosedinterval [a,b] andlet Nbeanynumberbetween f(a) and f(b), where f(a) ̸= f(b). Thenthereexistsanumber c in (a,b) suchthat f(c) = N.
. .x
.f(x)
.
.
.a .b
.f(a)
.f(b)
.N
.c
..
.c1
.
.c2
.
.c3
![Page 49: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/49.jpg)
. . . . . .
IllustratingtheIVTSupposethat f iscontinuousontheclosedinterval [a,b]
andlet Nbeanynumberbetween f(a) and f(b), where f(a) ̸= f(b). Thenthereexistsanumber c in (a,b) suchthat f(c) = N.
. .x
.f(x)
.
.
.a .b
.f(a)
.f(b)
.N
.c
..
.c1
.
.c2
.
.c3
![Page 50: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/50.jpg)
. . . . . .
IllustratingtheIVTSupposethat f iscontinuousontheclosedinterval [a,b]
andlet Nbeanynumberbetween f(a) and f(b), where f(a) ̸= f(b). Thenthereexistsanumber c in (a,b) suchthat f(c) = N.
. .x
.f(x)
.
.
.a .b
.f(a)
.f(b)
.N
.c
..
.c1
.
.c2
.
.c3
![Page 51: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/51.jpg)
. . . . . .
IllustratingtheIVTSupposethat f iscontinuousontheclosedinterval [a,b] andlet Nbeanynumberbetween f(a) and f(b), where f(a) ̸= f(b).
Thenthereexistsanumber c in (a,b) suchthat f(c) = N.
. .x
.f(x)
.
.
.a .b
.f(a)
.f(b)
.N
.c
..
.c1
.
.c2
.
.c3
![Page 52: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/52.jpg)
. . . . . .
IllustratingtheIVTSupposethat f iscontinuousontheclosedinterval [a,b] andlet Nbeanynumberbetween f(a) and f(b), where f(a) ̸= f(b). Thenthereexistsanumber c in (a,b) suchthat f(c) = N.
. .x
.f(x)
.
.
.a .b
.f(a)
.f(b)
.N
.c
.
.
.c1
.
.c2
.
.c3
![Page 53: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/53.jpg)
. . . . . .
IllustratingtheIVTSupposethat f iscontinuousontheclosedinterval [a,b] andlet Nbeanynumberbetween f(a) and f(b), where f(a) ̸= f(b). Thenthereexistsanumber c in (a,b) suchthat f(c) = N.
. .x
.f(x)
.
.
.a .b
.f(a)
.f(b)
.N
.c
..
.c1
.
.c2
.
.c3
![Page 54: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/54.jpg)
. . . . . .
IllustratingtheIVTSupposethat f iscontinuousontheclosedinterval [a,b] andlet Nbeanynumberbetween f(a) and f(b), where f(a) ̸= f(b). Thenthereexistsanumber c in (a,b) suchthat f(c) = N.
. .x
.f(x)
.
.
.a .b
.f(a)
.f(b)
.N
.c
.
.
.c1
.
.c2
.
.c3
![Page 55: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/55.jpg)
. . . . . .
WhattheIVT doesnotsay
TheIntermediateValueTheoremisan“existence”theorem.I Itdoesnotsayhowmanysuch c exist.I Italsodoesnotsayhowtofind c.
Still, itcanbeusediniterationorinconjunctionwithothertheoremstoanswerthesequestions.
![Page 56: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/56.jpg)
. . . . . .
UsingtheIVT
ExampleSupposeweareunawareofthesquarerootfunctionandthatit’scontinuous. Provethatthesquarerootoftwoexists.
Proof.Let f(x) = x2, acontinuousfunctionon [1, 2]. Note f(1) = 1 andf(2) = 4. Since 2 isbetween 1 and 4, thereexistsapoint c in(1, 2) suchthat
f(c) = c2 = 2.
Infact, wecan“narrowin”onthesquarerootof 2 by themethodofbisections.
![Page 57: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/57.jpg)
. . . . . .
UsingtheIVT
ExampleSupposeweareunawareofthesquarerootfunctionandthatit’scontinuous. Provethatthesquarerootoftwoexists.
Proof.Let f(x) = x2, acontinuousfunctionon [1, 2].
Note f(1) = 1 andf(2) = 4. Since 2 isbetween 1 and 4, thereexistsapoint c in(1, 2) suchthat
f(c) = c2 = 2.
Infact, wecan“narrowin”onthesquarerootof 2 by themethodofbisections.
![Page 58: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/58.jpg)
. . . . . .
UsingtheIVT
ExampleSupposeweareunawareofthesquarerootfunctionandthatit’scontinuous. Provethatthesquarerootoftwoexists.
Proof.Let f(x) = x2, acontinuousfunctionon [1, 2]. Note f(1) = 1 andf(2) = 4. Since 2 isbetween 1 and 4, thereexistsapoint c in(1, 2) suchthat
f(c) = c2 = 2.
Infact, wecan“narrowin”onthesquarerootof 2 by themethodofbisections.
![Page 59: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/59.jpg)
. . . . . .
UsingtheIVT
ExampleSupposeweareunawareofthesquarerootfunctionandthatit’scontinuous. Provethatthesquarerootoftwoexists.
Proof.Let f(x) = x2, acontinuousfunctionon [1, 2]. Note f(1) = 1 andf(2) = 4. Since 2 isbetween 1 and 4, thereexistsapoint c in(1, 2) suchthat
f(c) = c2 = 2.
Infact, wecan“narrowin”onthesquarerootof 2 by themethodofbisections.
![Page 60: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/60.jpg)
. . . . . .
Finding√2 bybisections
.
.x .f(x) = x2
..1 .1
..2 .4
..1.5 .2.25
..1.25 .1.5625
..1.375 .1.890625
..1.4375 .2.06640625
![Page 61: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/61.jpg)
. . . . . .
Finding√2 bybisections
.
.x .f(x) = x2
..1 .1
..2 .4
..1.5 .2.25
..1.25 .1.5625
..1.375 .1.890625
..1.4375 .2.06640625
![Page 62: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/62.jpg)
. . . . . .
Finding√2 bybisections
.
.x .f(x) = x2
..1 .1
..2 .4
..1.5 .2.25
..1.25 .1.5625
..1.375 .1.890625
..1.4375 .2.06640625
![Page 63: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/63.jpg)
. . . . . .
Finding√2 bybisections
.
.x .f(x) = x2
..1 .1
..2 .4
..1.5 .2.25
..1.25 .1.5625
..1.375 .1.890625
..1.4375 .2.06640625
![Page 64: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/64.jpg)
. . . . . .
Finding√2 bybisections
.
.x .f(x) = x2
..1 .1
..2 .4
..1.5 .2.25
..1.25 .1.5625
..1.375 .1.890625
..1.4375 .2.06640625
![Page 65: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/65.jpg)
. . . . . .
Finding√2 bybisections
.
.x .f(x) = x2
..1 .1
..2 .4
..1.5 .2.25
..1.25 .1.5625
..1.375 .1.890625
..1.4375 .2.06640625
![Page 66: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/66.jpg)
. . . . . .
UsingtheIVT
ExampleLet f(x) = x3 − x− 1. Showthatthereisazerofor f.
Solutionf(1) = −1 and f(2) = 5. Sothereisazerobetween 1 and 2.
(Morecarefulanalysisyields 1.32472.)
![Page 67: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/67.jpg)
. . . . . .
UsingtheIVT
ExampleLet f(x) = x3 − x− 1. Showthatthereisazerofor f.
Solutionf(1) = −1 and f(2) = 5. Sothereisazerobetween 1 and 2.(Morecarefulanalysisyields 1.32472.)
![Page 68: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/68.jpg)
. . . . . .
Outline
Continuity
TheIntermediateValueTheorem
BacktotheQuestions
![Page 69: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/69.jpg)
. . . . . .
BacktotheQuestions
TrueorFalseAtonepointinyourlifeyouwereexactlythreefeettall.
TrueorFalseAtonepointinyourlifeyourheightininchesequaledyourweightinpounds.
TrueorFalseRightnowtherearetwopointsonoppositesidesoftheEarthwithexactlythesametemperature.
![Page 70: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/70.jpg)
. . . . . .
Question1: True!
Let h(t) beheight, whichvariescontinuouslyovertime. Thenh(birth) < 3 ft and h(now) > 3 ft. Sothereisapoint c in(birth, now) where h(c) = 3.
![Page 71: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/71.jpg)
. . . . . .
BacktotheQuestions
TrueorFalseAtonepointinyourlifeyouwereexactlythreefeettall.
TrueorFalseAtonepointinyourlifeyourheightininchesequaledyourweightinpounds.
TrueorFalseRightnowtherearetwopointsonoppositesidesoftheEarthwithexactlythesametemperature.
![Page 72: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/72.jpg)
. . . . . .
Question2: True!
Let h(t) beheightininchesand w(t) beweightinpounds, bothvaryingcontinuouslyovertime. Let f(t) = h(t) −w(t). Formostofus(callyourmom), f(birth) > 0 and f(now) < 0. Sothereisapoint c in (birth, now) where f(c) = 0. Inotherwords,
h(c) −w(c) = 0 ⇐⇒ h(c) = w(c).
![Page 73: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/73.jpg)
. . . . . .
BacktotheQuestions
TrueorFalseAtonepointinyourlifeyouwereexactlythreefeettall.
TrueorFalseAtonepointinyourlifeyourheightininchesequaledyourweightinpounds.
TrueorFalseRightnowtherearetwopointsonoppositesidesoftheEarthwithexactlythesametemperature.
![Page 74: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/74.jpg)
. . . . . .
Question3
I Let T(θ) bethetemperatureatthepointontheequatoratlongitude θ.
I Howcanyouexpressthestatementthatthetemperatureonoppositesidesisthesame?
I Howcanyouensurethisistrue?
![Page 75: Lesson 5: Continuity](https://reader033.fdocuments.net/reader033/viewer/2022060119/558e302c1a28ab4d618b46af/html5/thumbnails/75.jpg)
. . . . . .
Question3: True!
I Let f(θ) = T(θ) − T(θ + 180◦)I Then
f(0) = T(0) − T(180)
whilef(180) = T(180) − T(360) = −f(0)
I Sosomewherebetween 0 and 180 thereisapoint θ wheref(θ) = 0!