Math 180
2.5 - Continuity
1
Informally, a function is continuous if you can
draw it without lifting up your pencil. Here are
some examples of continuous and discontinuous
functions:
2
continuous
3
continuous
4
discontinuous at 𝑥 = 2
5
discontinuous at 𝑥 = 3
6
Here’s the formal definition that we’ll be using:
𝑓(𝑥) is continuous at 𝒙 = 𝒄 if lim 𝑓(𝑥) = 𝑓(𝑐) .
𝑥→𝑐
(Note that 𝑓(𝑐) must be defined and lim 𝑓(𝑥) must exist.)
𝑥→𝑐
𝜋
2
ex: 𝑓 𝑥 = sin(𝑥) is continuous at 𝑥 = since
lim sin 𝑥 = ____ and 𝑓 𝜋/2 = ____.
𝑥→𝜋/2
7
Here’s the formal definition that we’ll be using:
𝑓(𝑥) is continuous at 𝒙 = 𝒄 if lim 𝑓(𝑥) = 𝑓(𝑐) .
𝑥→𝑐
(Note that 𝑓(𝑐) must be defined and lim 𝑓(𝑥) must exist.)
𝑥→𝑐
𝜋
2
ex: 𝑓 𝑥 = sin(𝑥) is continuous at 𝑥 = since
lim sin 𝑥 = ____ and 𝑓 𝜋/2 = ____.
𝑥→𝜋/2
8
Here’s the formal definition that we’ll be using:
𝑓(𝑥) is continuous at 𝒙 = 𝒄 if lim 𝑓(𝑥) = 𝑓(𝑐) .
𝑥→𝑐
(Note that 𝑓(𝑐) must be defined and lim 𝑓(𝑥) must exist.)
𝑥→𝑐
𝜋
2
ex: 𝑓 𝑥 = sin(𝑥) is continuous at 𝑥 = since
lim sin 𝑥 = ____ and 𝑓 𝜋/2 = ____.
𝑥→𝜋/2
9
Here’s the formal definition that we’ll be using:
𝑓(𝑥) is continuous at 𝒙 = 𝒄 if lim 𝑓(𝑥) = 𝑓(𝑐) .
𝑥→𝑐
(Note that 𝑓(𝑐) must be defined and lim 𝑓(𝑥) must exist.)
𝑥→𝑐
𝜋
2
ex: 𝑓 𝑥 = sin(𝑥) is continuous at 𝑥 = since
lim sin 𝑥 = ____
𝟏 and 𝑓 𝜋/2 = ____.
𝑥→𝜋/2
10
Here’s the formal definition that we’ll be using:
𝑓(𝑥) is continuous at 𝒙 = 𝒄 if lim 𝑓(𝑥) = 𝑓(𝑐) .
𝑥→𝑐
(Note that 𝑓(𝑐) must be defined and lim 𝑓(𝑥) must exist.)
𝑥→𝑐
𝜋
2
ex: 𝑓 𝑥 = sin(𝑥) is continuous at 𝑥 = since
lim sin 𝑥 = ____
𝟏 and 𝑓 𝜋/2 = ____.
𝟏
𝑥→𝜋/2
11
Here’s the formal definition that we’ll be using:
𝑓(𝑥) is continuous at 𝒙 = 𝒄 if lim 𝑓(𝑥) = 𝑓(𝑐) .
𝑥→𝑐
(Note that 𝑓(𝑐) must be defined and lim 𝑓(𝑥) must exist.)
𝑥→𝑐
𝜋
2
ex: 𝑓 𝑥 = sin(𝑥) is continuous at 𝑥 = since
lim sin 𝑥 = ____
𝟏 and 𝑓 𝜋/2 = ____.
𝟏
𝑥→𝜋/2
12
Here’s the formal definition that we’ll be using:
𝑓(𝑥) is continuous at 𝒙 = 𝒄 if lim 𝑓(𝑥) = 𝑓(𝑐) .
𝑥→𝑐
(Note that 𝑓(𝑐) must be defined and lim 𝑓(𝑥) must exist.)
𝑥→𝑐
𝜋
2
ex: 𝑓 𝑥 = sin(𝑥) is continuous at 𝑥 = since
lim sin 𝑥 = ____
𝟏 and 𝑓 𝜋/2 = ____.
𝟏
𝑥→𝜋/2
13
Here’s the formal definition that we’ll be using:
𝑓(𝑥) is continuous at 𝒙 = 𝒄 if lim 𝑓(𝑥) = 𝑓(𝑐) .
𝑥→𝑐
(Note that 𝑓(𝑐) must be defined and lim 𝑓(𝑥) must exist.)
𝑥→𝑐
𝜋
2
ex: 𝑓 𝑥 = sin(𝑥) is continuous at 𝑥 = since
lim sin 𝑥 = ____
𝟏 and 𝑓 𝜋/2 = ____.
𝟏
𝑥→𝜋/2
14
𝑓 𝑥 is continuous from the left at 𝒙 = 𝒄 if
lim− 𝑓 𝑥 = 𝑓 𝑐 .
𝑥→𝑐
𝑓 𝑥 is continuous from the right at 𝒙 = 𝒄 if
lim+ 𝑓 𝑥 = 𝑓 𝑐 .
𝑥→𝑐
ex: 𝑓 𝑥 = 𝑥 − 1 is continuous from the right at 𝑥 =
1 since lim+ 𝑥 − 1 = ____ and 𝑓 1 = ____.
𝑥→1
15
𝑓 𝑥 is continuous from the left at 𝒙 = 𝒄 if
lim− 𝑓 𝑥 = 𝑓 𝑐 .
𝑥→𝑐
𝑓 𝑥 is continuous from the right at 𝒙 = 𝒄 if
lim+ 𝑓 𝑥 = 𝑓 𝑐 .
𝑥→𝑐
ex: 𝑓 𝑥 = 𝑥 − 1 is continuous from the right at 𝑥 =
1 since lim+ 𝑥 − 1 = ____ and 𝑓 1 = ____.
𝑥→1
16
𝑓 𝑥 is continuous from the left at 𝒙 = 𝒄 if
lim− 𝑓 𝑥 = 𝑓 𝑐 .
𝑥→𝑐
𝑓 𝑥 is continuous from the right at 𝒙 = 𝒄 if
lim+ 𝑓 𝑥 = 𝑓 𝑐 .
𝑥→𝑐
ex: 𝑓 𝑥 = 𝑥 − 1 is continuous from the right at
𝑥 = 1 since lim+ 𝑥 − 1 = ____ and 𝑓 1 = ____.
𝑥→1
17
𝑓 𝑥 is continuous from the left at 𝒙 = 𝒄 if
lim− 𝑓 𝑥 = 𝑓 𝑐 .
𝑥→𝑐
𝑓 𝑥 is continuous from the right at 𝒙 = 𝒄 if
lim+ 𝑓 𝑥 = 𝑓 𝑐 .
𝑥→𝑐
ex: 𝑓 𝑥 = 𝑥 − 1 is continuous from the right at
𝟎 and 𝑓 1 = ____.
𝑥 = 1 since lim+ 𝑥 − 1 = ____
𝑥→1
18
𝑓 𝑥 is continuous from the left at 𝒙 = 𝒄 if
lim− 𝑓 𝑥 = 𝑓 𝑐 .
𝑥→𝑐
𝑓 𝑥 is continuous from the right at 𝒙 = 𝒄 if
lim+ 𝑓 𝑥 = 𝑓 𝑐 .
𝑥→𝑐
ex: 𝑓 𝑥 = 𝑥 − 1 is continuous from the right at
𝟎 and 𝑓 1 = ____.
𝟎
𝑥 = 1 since lim+ 𝑥 − 1 = ____
𝑥→1
19
𝑓 𝑥 is continuous from the left at 𝒙 = 𝒄 if
lim− 𝑓 𝑥 = 𝑓 𝑐 .
𝑥→𝑐
𝑓 𝑥 is continuous from the right at 𝒙 = 𝒄 if
lim+ 𝑓 𝑥 = 𝑓 𝑐 .
𝑥→𝑐
ex: 𝑓 𝑥 = 𝑥 − 1 is continuous from the right at
𝟎 and 𝑓 1 = ____.
𝟎
𝑥 = 1 since lim+ 𝑥 − 1 = ____
𝑥→1
20
𝑓 𝑥 is continuous from the left at 𝒙 = 𝒄 if
lim− 𝑓 𝑥 = 𝑓 𝑐 .
𝑥→𝑐
𝑓 𝑥 is continuous from the right at 𝒙 = 𝒄 if
lim+ 𝑓 𝑥 = 𝑓 𝑐 .
𝑥→𝑐
ex: 𝑓 𝑥 = 𝑥 − 1 is continuous from the right at
𝟎 and 𝑓 1 = ____.
𝟎
𝑥 = 1 since lim+ 𝑥 − 1 = ____
𝑥→1
21
𝑓(𝑥) is continuous on an interval if it is
continuous at every point of the interval.
1
𝑥
ex: 𝑓 𝑥 = is continuous on (0, ∞).
22
𝑓(𝑥) is continuous on an interval if it is
continuous at every point of the interval.
1
𝑥
ex: 𝑓 𝑥 = is continuous on (0, ∞).
23
Ex 1.
Is 𝑓 continuous or discontinuous at each 𝑥-value? Is 𝑓
continuous from the left? Is 𝑓 continuous from the
right?
𝑥=2
𝑥=3
𝑥=4
24
Ex 2.
𝑥−1
2
Explain why 𝑓 𝑥 =
3−𝑥
discontinuous at 𝑥 = 0.
if 𝑥 < 0 is
if 𝑥 ≥ 0
25
Ex 2.
𝑥−1
2
Explain why 𝑓 𝑥 =
3−𝑥
discontinuous at 𝑥 = 0.
if 𝑥 < 0 is
if 𝑥 ≥ 0
26
Ex 3.
How would you define 𝑓 2 in a way that makes
𝑓 𝑥 =
𝑥 2 −5𝑥+6
𝑥−2
continuous at 𝑥 = 2?
(This is called “removing the discontinuity”.)
27
Ex 3.
How would you define 𝑓 2 in a way that makes
𝑓 𝑥 =
𝑥 2 −5𝑥+6
𝑥−2
continuous at 𝑥 = 2?
(This is called “removing the discontinuity”.)
28
Ex 3.
How would you define 𝑓 2 in a way that makes
𝑓 𝑥 =
𝑥 2 −5𝑥+6
𝑥−2
continuous at 𝑥 = 2?
(This is called “removing the discontinuity”.)
29
Properties of Continuous Functions
If 𝑓 and 𝑔 are continuous at 𝑥 = 𝑐, then all of
the following functions are also continuous at
𝑥 = 𝑐:
𝑓+𝑔
𝑓⋅𝑔
𝑓−𝑔
𝑓 𝑔
𝑘⋅𝑓
𝑓
𝑛
𝑛
𝑓
30
Why? Here’s the proof of 𝑓 + 𝑔:
lim 𝑓 + 𝑔 𝑥 = lim 𝑓 𝑥 + 𝑔 𝑥
𝑥→𝑐
𝑥→𝑐
= lim 𝑓(𝑥) + lim 𝑔(𝑥)
𝑥→𝑐
𝑥→𝑐
=𝑓 𝑐 +𝑔 𝑐
= (𝑓 + 𝑔)(𝑐)
31
Notes:
• Any polynomial 𝑃 𝑥 is continuous since,
lim 𝑃 𝑥 = 𝑃(𝑐).
𝑥→𝑐
ex: 𝑓 𝑥 = 𝑥 2 + 3𝑥 − 4 is continuous
everywhere (that is, (−∞, ∞)).
• Any rational function 𝑃(𝑥)/𝑄(𝑥) is continuous
wherever 𝑄 𝑥 ≠ 0.
ex: 𝑓 𝑥 =
𝑥+1
is
𝑥−2
continuous for all 𝑥 except 2.
32
Notes:
• Any polynomial 𝑃 𝑥 is continuous since,
lim 𝑃 𝑥 = 𝑃(𝑐).
𝑥→𝑐
ex: 𝑓 𝑥 = 𝑥 2 + 3𝑥 − 4 is continuous
everywhere (that is, (−∞, ∞)).
• Any rational function 𝑃(𝑥)/𝑄(𝑥) is continuous
wherever 𝑄 𝑥 ≠ 0.
ex: 𝑓 𝑥 =
𝑥+1
is
𝑥−2
continuous for all 𝑥 except 2.
33
•
𝑥 , sin 𝑥, and cos 𝑥 are continuous
________.
•
𝑥 is continuous ____________.
34
•
𝑥 , sin 𝑥, and cos 𝑥 are continuous
for all 𝒙
________.
•
𝑥 is continuous ____________.
35
•
𝑥 , sin 𝑥, and cos 𝑥 are continuous
for all 𝒙
________.
•
𝑥 is continuous ____________.
36
•
𝑥 , sin 𝑥, and cos 𝑥 are continuous
for all 𝒙
________.
•
for all 𝒙 ≥ 𝟎
𝑥 is continuous ____________.
37
• If 𝑓 is continuous at 𝑐 and 𝑔 is continuous at
𝑓(𝑐), then 𝑔 ∘ 𝑓 is continuous at 𝑐.
ex: 𝑥 + 1 is continuous everywhere on its
domain since 𝑓 𝑥 = 𝑥 + 1 is continuous and
𝑔 𝑥 = 𝑥 is continuous, so 𝑔 ∘ 𝑓 𝑥 =
𝑔 𝑓 𝑥 = 𝑥 + 1 is continuous.
38
• If 𝑓 is continuous at 𝑐 and 𝑔 is continuous at
𝑓(𝑐), then 𝑔 ∘ 𝑓 is continuous at 𝑐.
ex: 𝑥 + 1 is continuous everywhere on its
domain since 𝑓 𝑥 = 𝑥 + 1 is continuous and
𝑔 𝑥 = 𝑥 is continuous, so 𝑔 ∘ 𝑓 𝑥 =
𝑔 𝑓 𝑥 = 𝑥 + 1 is continuous.
39
Ex 4.
𝑥+3
At what points is the function 𝑦 = 2
𝑥 −3𝑥−10
continuous?
Ex 5.
At what points is the function 𝑦 = sec 2𝑥
continuous?
40
One useful fact about continuous functions is
that they satisfy the
_________________________.
A function has this property if it takes on all
values between any two function values.
41
One useful fact about continuous functions is
that they satisfy the
Intermediate Value Property
_________________________.
A function has this property if it takes on all
values between any two function values.
42
Intermediate Value Theorem
If 𝑓 is a continuous function on a closed interval
𝑎, 𝑏 , and if 𝑦0 is any value between 𝑓(𝑎) and
𝑓(𝑏), then 𝑦0 = 𝑓(𝑐) for some 𝑐 in 𝑎, 𝑏 .
43
Ex 6.
Use the Intermediate Value Theorem to show that
there is a root of 3 𝑥 = 1 − 𝑥 in the interval 0,1 .
44
Ex 6.
Use the Intermediate Value Theorem to show that
there is a root of 3 𝑥 = 1 − 𝑥 in the interval 0,1 .
45
Ex 6.
Use the Intermediate Value Theorem to show that
there is a root of 3 𝑥 = 1 − 𝑥 in the interval 0,1 .
46
Ex 6.
Use the Intermediate Value Theorem to show that
there is a root of 3 𝑥 = 1 − 𝑥 in the interval 0,1 .
47