Continuous function have an important property that small changes in input do not produce large changes in output. The Intermediate Value Theorem shows that a continuous function takes all values between any two values. From this we know that your height and weight were once the same, and right now there are two points on opposite sides of the world with the same temperature!

1. Section 1.5
Continuity
V63.0121.006/016, Calculus I
February 2, 2010
Announcements
Office Hours: M,W 1:30–2:30, R 9–10 (CIWW 726)
Written Assignment #2 due Thursday
First Quiz: Friday February 12 in recitation (§§1.1–1.4)
. . . . . .

2. Hatsumon
Here are some discussion questions to start.
True or False
At some point in your life you were exactly three feet tall.
. . . . . .

3. Hatsumon
Here are some discussion questions to start.
True or False
At some point in your life you were exactly three feet tall.
True or False
At some point in your life your height (in inches) was equal to
your weight (in pounds).
. . . . . .

4. Hatsumon
Here are some discussion questions to start.
True or False
At some point in your life you were exactly three feet tall.
True or False
At some point in your life your height (in inches) was equal to
your weight (in pounds).
True or False
Right now there are a pair of points on opposite sides of the
world measuring the exact same temperature.
. . . . . .

5. Outline
Continuity
The Intermediate Value Theorem
Back to the Questions
. . . . . .

6. Recall: Direct Substitution Property
Theorem (The Direct Substitution Property)
If f is a polynomial or a rational function and a is in the domain of
f, then
lim f(x) = f(a)
x →a
This property is so useful it’s worth naming.
. . . . . .

7. Definition of Continuity
Definition
Let f be a function
defined near a. We say
that f is continuous at a
if
lim f(x) = f(a).
x →a
. . . . . .

8. Definition of Continuity
Definition y
.
Let f be a function
defined near a. We say
that f is continuous at a f
.(a ) .
if
lim f(x) = f(a).
x →a
A function f is
continuous if it is
continuous at every . x
.
point in its domain. a
.
. . . . . .

9. Scholium
Definition
Let f be a function defined near a. We say that f is continuous at
a if
lim f(x) = f(a).
x→a
There are three important parts to this definition.
The function has to have a limit at a,
the function has to have a value at a,
and these values have to agree.
. . . . . .

10. Free Theorems
Theorem
(a) Any polynomial is continuous everywhere; that is, it is
continuous on R = (−∞, ∞).
(b) Any rational function is continuous wherever it is defined;
that is, it is continuous on its domain.
. . . . . .

11. Showing a function is continuous
Example
√
Let f(x) = 4x + 1. Show that f is continuous at 2.
. . . . . .

12. Showing a function is continuous
Example
√
Let f(x) = 4x + 1. Show that f is continuous at 2.
Solution
We want to show that lim f(x) = f(2). We have
x →2
√ √ √
lim f(x) = lim 4x + 1 = lim (4x + 1) = 9 = 3 = f(2).
x →a x →2 x →2
Each step comes from the limit laws.
. . . . . .

13. Showing a function is continuous
Example
√
Let f(x) = 4x + 1. Show that f is continuous at 2.
Solution
We want to show that lim f(x) = f(2). We have
x →2
√ √ √
lim f(x) = lim 4x + 1 = lim (4x + 1) = 9 = 3 = f(2).
x →a x →2 x →2
Each step comes from the limit laws.
Question
At which other points is f continuous?
. . . . . .

14. At which other points?
√
For reference: f(x) = 4x + 1
If a > −1/4, then lim (4x + 1) = 4a + 1 > 0, so
x →a
√ √ √
lim f(x) = lim 4x + 1 = lim (4x + 1) = 4a + 1 = f(a)
x→a x →a x →a
and f is continuous at a.
. . . . . .

15. At which other points?
√
For reference: f(x) = 4x + 1
If a > −1/4, then lim (4x + 1) = 4a + 1 > 0, so
x →a
√ √ √
lim f(x) = lim 4x + 1 = lim (4x + 1) = 4a + 1 = f(a)
x→a x →a x →a
and f is continuous at a.
If a = −1/4, then 4x + 1 < 0 to the left of a, which means
√
4x + 1 is undefined. Still,
√ √ √
lim f(x) = lim 4x + 1 = lim (4x + 1) = 0 = 0 = f(a)
x →a + x→a+ x →a +
so f is continuous on the right at a = −1/4.
. . . . . .

16. Showing a function is continuous
Example
√
Let f(x) = 4x + 1. Show that f is continuous at 2.
Solution
We want to show that lim f(x) = f(2). We have
x →2
√ √ √
lim f(x) = lim 4x + 1 = lim (4x + 1) = 9 = 3 = f(2).
x →a x →2 x →2
Each step comes from the limit laws.
Question
At which other points is f continuous?
Answer
The function f is continuous on (−1/4, ∞).
. . . . . .

17. Showing a function is continuous
Example
√
Let f(x) = 4x + 1. Show that f is continuous at 2.
Solution
We want to show that lim f(x) = f(2). We have
x →2
√ √ √
lim f(x) = lim 4x + 1 = lim (4x + 1) = 9 = 3 = f(2).
x →a x →2 x →2
Each step comes from the limit laws.
Question
At which other points is f continuous?
Answer
The function f is continuous on (−1/4, ∞). It is right continuous
at −1/4 since lim f(x) = f(−1/4).
x→−1/4+
. . . . . .

18. The Limit Laws give Continuity Laws
Theorem
If f(x) and g(x) are continuous at a and c is a constant, then the
following functions are also continuous at a:
(f + g)(x)
(f − g)(x)
(cf)(x)
(fg)(x)
f
(x) (if g(a) ̸= 0)
g
. . . . . .

19. Why a sum of continuous functions is continuous
We want to show that
lim (f + g)(x) = (f + g)(a).
x →a
We just follow our nose:
lim (f + g)(x) = lim [f(x) + g(x)] (def of f + g)
x →a x →a
= lim f(x) + lim g(x) (if these limits exist)
x →a x →a
= f(a) + g(a) (they do; f and g are cts.)
= (f + g)(a) (def of f + g again)
. . . . . .

20. Trigonometric functions are continuous
sin and cos are continuous
on R.
.
s
. in
. . . . . .

21. Trigonometric functions are continuous
sin and cos are continuous
on R.
c
. os
.
s
. in
. . . . . .

22. Trigonometric functions are continuous
t
.an
sin and cos are continuous
on R.
sin 1
tan = and sec =
cos cos c
. os
are continuous on their
domain, which is
{π } .
R + kπ k ∈ Z . s
. in
2
. . . . . .

23. Trigonometric functions are continuous
t
.an s
. ec
sin and cos are continuous
on R.
sin 1
tan = and sec =
cos cos c
. os
are continuous on their
domain, which is
{π } .
R + kπ k ∈ Z . s
. in
2
. . . . . .

24. Trigonometric functions are continuous
t
.an s
. ec
sin and cos are continuous
on R.
sin 1
tan = and sec =
cos cos c
. os
are continuous on their
domain, which is
{π } .
R + kπ k ∈ Z . s
. in
2
cos 1
cot = and csc =
sin sin
are continuous on their
domain, which is
R { k π | k ∈ Z }.
c
. ot
. . . . . .

25. Trigonometric functions are continuous
t
.an s
. ec
sin and cos are continuous
on R.
sin 1
tan = and sec =
cos cos c
. os
are continuous on their
domain, which is
{π } .
R + kπ k ∈ Z . s
. in
2
cos 1
cot = and csc =
sin sin
are continuous on their
domain, which is
R { k π | k ∈ Z }.
c
. ot . sc
c
. . . . . .

26. Exponential and Logarithmic functions are continuous
For any base a > 1, .x
a
the function x → ax is
continuous on R
.
. . . . . .

27. Exponential and Logarithmic functions are continuous
For any base a > 1, .x
a
the function x → ax is .oga x
l
continuous on R
the function loga is
continuous on its .
domain: (0, ∞)
. . . . . .

28. Exponential and Logarithmic functions are continuous
For any base a > 1, .x
a
the function x → ax is .oga x
l
continuous on R
the function loga is
continuous on its .
domain: (0, ∞)
In particular ex and
ln = loge are continuous
on their domains
. . . . . .

29. Inverse trigonometric functions are mostly continuous
sin−1 and cos−1 are continuous on (−1, 1), left continuous
at 1, and right continuous at −1.
.
π
.
. /2
π
.
. in−1
s
.
−
. π/2
. . . . . .

30. Inverse trigonometric functions are mostly continuous
sin−1 and cos−1 are continuous on (−1, 1), left continuous
at 1, and right continuous at −1.
.
.
π
. os−1
c .
. /2
π
. .
. in−1
s
.
−
. π/2
. . . . . .

31. Inverse trigonometric functions are mostly continuous
sin−1 and cos−1 are continuous on (−1, 1), left continuous
at 1, and right continuous at −1.
sec−1 and csc−1 are continuous on (−∞, −1) ∪ (1, ∞), left
continuous at −1, and right continuous at 1.
.
.
π
. os−1
c . . ec−1
s
. /2
π
. .
. in−1
s
.
−
. π/2
. . . . . .

32. Inverse trigonometric functions are mostly continuous
sin−1 and cos−1 are continuous on (−1, 1), left continuous
at 1, and right continuous at −1.
sec−1 and csc−1 are continuous on (−∞, −1) ∪ (1, ∞), left
continuous at −1, and right continuous at 1.
.
.
π
. os−1
c . . ec−1
s
. /2
π
. sc−1
c
. .
. in−1
s
.
−
. π/2
. . . . . .

33. Inverse trigonometric functions are mostly continuous
sin−1 and cos−1 are continuous on (−1, 1), left continuous
at 1, and right continuous at −1.
sec−1 and csc−1 are continuous on (−∞, −1) ∪ (1, ∞), left
continuous at −1, and right continuous at 1.
tan−1 and cot−1 are continuous on R.
.
.
π
. os−1
c . . ec−1
s
. /2
π
.an−1
t
. sc−1
c
. .
. in−1
s
.
−
. π/2
. . . . . .

34. Inverse trigonometric functions are mostly continuous
sin−1 and cos−1 are continuous on (−1, 1), left continuous
at 1, and right continuous at −1.
sec−1 and csc−1 are continuous on (−∞, −1) ∪ (1, ∞), left
continuous at −1, and right continuous at 1.
tan−1 and cot−1 are continuous on R.
.
.
π
. ot−1
c
. os−1
c . . ec−1
s
. /2
π
.an−1
t
. sc−1
c
. .
. in−1
s
.
−
. π/2
. . . . . .

35. What could go wrong?
In what ways could a function f fail to be continuous at a point a?
Look again at the definition:
lim f(x) = f(a)
x →a
. . . . . .

36. Continuity FAIL
Example
Let {
x2 if 0 ≤ x ≤ 1
f(x ) =
2x if 1 < x ≤ 2
At which points is f continuous?
. . . . . .

37. Continuity FAIL: The limit does not exist
Example
Let {
x2 if 0 ≤ x ≤ 1
f(x ) =
2x if 1 < x ≤ 2
At which points is f continuous?
Solution
At any point a in [0, 2] besides 1, lim f(x) = f(a) because f is
x →a
represented by a polynomial near a, and polynomials have the
direct substitution property. However,
lim f(x) = lim x2 = 12 = 1
x →1 − x →1 −
lim f(x) = lim 2x = 2(1) = 2
x→1+ x →1 +
So f has no limit at 1. Therefore f is not continuous at 1.
. . . . . .

38. Graphical Illustration of Pitfall #1
y
.
. .
4 .
. .
3
. .
2 .
. .
1 .
. . . . x
.
−
. 1 1
. 2
.
. 1 .
−
. . . . . .

39. Continuity FAIL
Example
Let
x2 + 2x + 1
f(x ) =
x+1
At which points is f continuous?
. . . . . .

40. Continuity FAIL: The function has no value
Example
Let
x2 + 2x + 1
f(x ) =
x+1
At which points is f continuous?
Solution
Because f is rational, it is continuous on its whole domain. Note
that −1 is not in the domain of f, so f is not continuous there.
. . . . . .

41. Graphical Illustration of Pitfall #2
y
.
. .
1
. . x
.
−
. 1
f cannot be continuous where it has no value.
. . . . . .

42. Continuity FAIL
Example
Let {
7 if x ̸= 1
f(x) =
π if x = 1
At which points is f continuous?
. . . . . .

43. Continuity FAIL: function value ̸= limit
Example
Let {
7 if x ̸= 1
f(x) =
π if x = 1
At which points is f continuous?
Solution
f is not continuous at 1 because f(1) = π but lim f(x) = 7.
x →1
. . . . . .

44. Graphical Illustration of Pitfall #3
y
.
. .
7 .
. .
π .
. . x
.
1
.
. . . . . .

45. Special types of discontinuites
removable discontinuity The limit lim f(x) exists, but f is not
x →a
defined at a or its value at a is not equal to the limit
at a.
jump discontinuity The limits lim f(x) and lim f(x) exist, but
x →a − x→a+
are different.
. . . . . .

46. Graphical representations of discontinuities
.
y
.
y
.
. .
7 .
. .
2 .
. .
π .
. .
1 .
. . x
. . . x
.
1
.
1
.
removable
jump
. . . . . .

47. Graphical representations of discontinuities
.
y
.
y
.
P
. resto! continuous!
. .
7 .
. .
2 .
. .
π
. .
1 .
. . x
. . . x
.
1
.
1
.
removable
jump
. . . . . .

48. Graphical representations of discontinuities
.
y
.
y
.
P
. resto! continuous!
. .
7 .
. .
2 .
. .
π
. .
1 . . ontinuous?
c
. . x
. . . x
.
1
.
1
.
removable
jump
. . . . . .

49. Graphical representations of discontinuities
.
y
.
y
.
P
. resto! continuous!
. .
7 .
. .
2 . . ontinuous?
c
. .
π
. .
1 .
. . x
. . . x
.
1
.
1
.
removable
jump
. . . . . .

50. Graphical representations of discontinuities
.
y
.
y
.
P
. resto! continuous!
. .
7 .
. .
2 .
. . ontinuous?
c
. .
π
. .
1 .
. . x
. . . x
.
1
.
1
.
removable
jump
. . . . . .

51. Special types of discontinuites
removable discontinuity The limit lim f(x) exists, but f is not
x →a
defined at a or its value at a is not equal to the limit
at a. By re-defining f(a) = lim f(x), f can be made
x →a
continuous at a
jump discontinuity The limits lim f(x) and lim f(x) exist, but
x →a − x→a+
are different.
. . . . . .

52. Special types of discontinuites
removable discontinuity The limit lim f(x) exists, but f is not
x →a
defined at a or its value at a is not equal to the limit
at a. By re-defining f(a) = lim f(x), f can be made
x →a
continuous at a
jump discontinuity The limits lim f(x) and lim f(x) exist, but
x →a − x→a+
are different. The function cannot be made
continuous by changing a single value.
. . . . . .

53. The greatest integer function
[[x]] is the greatest integer ≤ x.
y
.
. .
3
x [[x]] y
. = [[x]]
0 0 . .
2 . .
1 1
1.5 1 . .
1 . .
1.9 1
2.1 2 . . . . . . x
.
−0.5 −1 −
. 2 −
. 1 1
. 2
. 3
.
−0.9 −1 .. 1 .
−
−1.1 −2
. .. 2 .
−
. . . . . .

54. The greatest integer function
[[x]] is the greatest integer ≤ x.
y
.
. .
3
x [[x]] y
. = [[x]]
0 0 . .
2 . .
1 1
1.5 1 . .
1 . .
1.9 1
2.1 2 . . . . . . x
.
−0.5 −1 −
. 2 −
. 1 1
. 2
. 3
.
−0.9 −1 .. 1 .
−
−1.1 −2
. .. 2 .
−
This function has a jump discontinuity at each integer.
. . . . . .

55. Outline
Continuity
The Intermediate Value Theorem
Back to the Questions
. . . . . .

56. A Big Time Theorem
Theorem (The Intermediate Value Theorem)
Suppose that f is continuous on the closed interval [a, b] and let N
be any number between f(a) and f(b), where f(a) ̸= f(b). Then
there exists a number c in (a, b) such that f(c) = N.
. . . . . .

57. Illustrating the IVT
f
.(x )
. x
.
. . . . . .

58. Illustrating the IVT
Suppose that f is continuous on the closed interval [a, b]
f
.(x )
.
.
. x
.
. . . . . .

59. Illustrating the IVT
Suppose that f is continuous on the closed interval [a, b]
f
.(x )
f
.(b ) .
f
.(a ) .
. a
. x
.
b
.
. . . . . .

60. Illustrating the IVT
Suppose that f is continuous on the closed interval [a, b] and let N
be any number between f(a) and f(b), where f(a) ̸= f(b).
f
.(x )
f
.(b ) .
N
.
f
.(a ) .
. a
. x
.
b
.
. . . . . .

61. Illustrating the IVT
Suppose that f is continuous on the closed interval [a, b] and let N
be any number between f(a) and f(b), where f(a) ̸= f(b). Then
there exists a number c in (a, b) such that f(c) = N.
f
.(x )
f
.(b ) .
N
. .
f
.(a ) .
. a
. c
. x
.
b
.
. . . . . .

62. Illustrating the IVT
Suppose that f is continuous on the closed interval [a, b] and let N
be any number between f(a) and f(b), where f(a) ̸= f(b). Then
there exists a number c in (a, b) such that f(c) = N.
f
.(x )
f
.(b ) .
N
.
f
.(a ) .
. a
. x
.
b
.
. . . . . .

63. Illustrating the IVT
Suppose that f is continuous on the closed interval [a, b] and let N
be any number between f(a) and f(b), where f(a) ̸= f(b). Then
there exists a number c in (a, b) such that f(c) = N.
f
.(x )
f
.(b ) .
N
. . . .
f
.(a ) .
. a c
. .1 x
.
c
.2 c b
.3 .
. . . . . .

64. What the IVT does not say
The Intermediate Value Theorem is an “existence” theorem.
It does not say how many such c exist.
It also does not say how to find c.
Still, it can be used in iteration or in conjunction with other
theorems to answer these questions.
. . . . . .

65. Using the IVT
Example
Suppose we are unaware of the square root function and that it’s
continuous. Prove that the square root of two exists.
. . . . . .

66. Using the IVT
Example
Suppose we are unaware of the square root function and that it’s
continuous. Prove that the square root of two exists.
Proof.
Let f(x) = x2 , a continuous function on [1, 2].
. . . . . .

67. Using the IVT
Example
Suppose we are unaware of the square root function and that it’s
continuous. Prove that the square root of two exists.
Proof.
Let f(x) = x2 , a continuous function on [1, 2]. Note f(1) = 1 and
f(2) = 4. Since 2 is between 1 and 4, there exists a point c in
(1, 2) such that
f(c) = c2 = 2.
. . . . . .

68. Using the IVT
Example
Suppose we are unaware of the square root function and that it’s
continuous. Prove that the square root of two exists.
Proof.
Let f(x) = x2 , a continuous function on [1, 2]. Note f(1) = 1 and
f(2) = 4. Since 2 is between 1 and 4, there exists a point c in
(1, 2) such that
f(c) = c2 = 2.
In fact, we can “narrow in” on the square root of 2 by the method
of bisections.
. . . . . .

69. √
Finding 2 by bisections
. .(x) = x2
x f
. ..
2 4
. ..
1 1
. . . . . .

70. √
Finding 2 by bisections
. .(x) = x2
x f
. ..
2 4
. ..
1 1
. . . . . .

71. √
Finding 2 by bisections
. .(x) = x2
x f
. ..
2 4
. .5 . . .25
1 2
. ..
1 1
. . . . . .

72. √
Finding 2 by bisections
. .(x) = x2
x f
. ..
2 4
. .5 . . .25
1 2
. .25 . . .5625
1 1
. ..
1 1
. . . . . .

73. √
Finding 2 by bisections
. .(x) = x2
x f
. ..
2 4
1
. .5 . . .25
2
1
. .375 . . .890625
1
1
. .25 . . .5625
1
. ..
1 1
. . . . . .

74. √
Finding 2 by bisections
. .(x) = x2
x f
. ..
2 4
. . .25
1
. .5 . 2.06640625
1
. .4375
1
. .375 . . .890625
2
1
.
1
. .25 . . .5625
1
. ..
1 1
. . . . . .

75. Using the IVT
Example
Let f(x) = x3 − x − 1. Show that there is a zero for f.
Solution
f(1) = −1 and f(2) = 5. So there is a zero between 1 and 2.
. . . . . .

76. Using the IVT
Example
Let f(x) = x3 − x − 1. Show that there is a zero for f.
Solution
f(1) = −1 and f(2) = 5. So there is a zero between 1 and 2.
(More careful analysis yields 1.32472.)
. . . . . .

77. Outline
Continuity
The Intermediate Value Theorem
Back to the Questions
. . . . . .

78. Back to the Questions
True or False
At one point in your life you were exactly three feet tall.
. . . . . .

79. Question 1: True!
Let h(t) be height, which varies continuously over time.
Then h(birth) < 3 ft and h(now) > 3 ft.
So by the IVT there is a point c in (birth, now) where
h(c) = 3.
. . . . . .

80. Back to the Questions
True or False
At one point in your life you were exactly three feet tall.
True or False
At one point in your life your height in inches equaled your
weight in pounds.
. . . . . .

81. Question 2: True!
Let h(t) be height in inches and w(t) be weight in pounds,
both varying continuously over time.
Let f(t) = h(t) − w(t).
For most of us (call your mom), f(birth) > 0 and f(now) < 0.
So by the IVT there is a point c in (birth, now) where
f(c) = 0.
In other words,
h(c) − w(c) = 0 ⇐⇒ h(c) = w(c).
. . . . . .

82. Back to the Questions
True or False
At one point in your life you were exactly three feet tall.
True or False
At one point in your life your height in inches equaled your
weight in pounds.
True or False
Right now there are two points on opposite sides of the Earth
with exactly the same temperature.
. . . . . .

83. Question 3
Let T(θ) be the temperature at the point on the equator at
longitude θ.
How can you express the statement that the temperature on
opposite sides is the same?
How can you ensure this is true?
. . . . . .

84. Question 3: True!
Let f(θ) = T(θ) − T(θ + 180◦ )
Then
f(0) = T(0) − T(180)
while
f(180) = T(180) − T(360) = −f(0)
So somewhere between 0 and 180 there is a point θ where
f(θ) = 0!
. . . . . .

85. What have we learned today?
Definition: a function is continuous at a point if the limit of
the function at that point agrees with the value of the
function at that point.
We often make a fundamental assumption that functions we
meet in nature are continuous.
The Intermediate Value Theorem is a basic property of real
numbers that we need and use a lot.
. . . . . .