CONTINUITY.pdf

C O N T I N U I T Y
TOPIC 8
C A L C U L U S I W I T H A N A L Y T I C G E O M E T R Y

S U B T O P I C S
❑ Definition of Continuity
❑ Types of Continuity and Discontinuity
❑ Continuity of a Composite Function
❑ Continuity Interval

Overview
Many functions have the property that their graphs can be traced with a pencil
without lifting the pencil from the page. Such functions are called continuous. The
property of continuity Is exhibited by various aspects of nature. The water flow in
the river is continuous. The flow of time in human life is continuous, you are
getting older continuously. And so on. Similarly, in Mathematics, we have the
notion of continuity of a function.
What it simply mean is that a functions is said to be continuous of you can sketch its
curve on a graph without lifting your pen even once. It is a very straight forward
and close to accurate definition actually. But for the sake of higher mathematics,
we must define it in a more precise

Overview
● The limit of a function as x approaches a can often be found simply by
calculating the value of the function at a. Functions with this property
are called continuous at a.
● We will see that the mathematical definition of continuity corresponds
closely with the meaning of the word continuity in everyday language.
(A continuous process is one that takes place gradually, without
interruption or abrupt change.)

Overview
● In fact, the change in f (x) can be kept as small as we please by keeping
the change in x sufficiently small.
● If f is defined near a (in other words, f is defined on an open interval
containing a, except perhaps at a), we say that f is discontinuous at a
(or f has a discontinuity at a) if f is not continuous at a.
● Physical phenomena are usually continuous. For instance, the
displacement or velocity of a vehicle varies continuously with time, as
does a person’s height. But discontinuities do occur in such situations
as electric currents.

Overview
● Geometrically, you can think of a function that is continuous at every
number in an interval as a function whose graph has no break in it. The
graph can be drawn without removing your pen from the paper.

DEFINITION OF CONTINUITY
A function f is continuous at x if it satisfies the following condition:
Definition of Continuity
A function f is continuous at x=a when:
1. 𝟏. 𝒇 𝒂 𝐢𝐬 𝐝𝐞𝐟𝐢𝐧𝐞𝐝
2. 2. 𝐥𝐢𝐦
𝒙→𝒂
𝒇 𝒙 𝒆𝒙𝒊𝒔𝒕
3. 3. 𝐥𝐢𝐦
𝒙→𝒂
𝒇 𝒙 = 𝒇(𝒂)
If any one of the condition is not met, the
function in not continuous at x=a
The definition says that f is continuous at a if f (x)
approaches f (a) as x approaches a. Thus a continuous
function f has the property that a small change in x
produces only a small change in f (x).

Illustrative Examples
lim
𝑥→1
𝑓(𝑥) = 𝑥2
+ 𝑥 + 1
Checking of Continuity
Is the function defined at x = 1? Yes
Does the limit of the function
as x approaches 1 exist?
Yes
as x approaches 0 equal the function
value at x = 1?
Yes

lim
𝑥→1
𝑓(𝑥) = 𝑥2 + 𝑥 + 1
Checking of
Continuity
Is the function
defined at x = 1?
Yes
Does the limit of
the function
as x approaches 1
exist?
Yes
Does the limit of
the function
as x approaches 1
equal the function
value at x = 1?
Yes
x 0.9 0.99 0.999 1 1.01 1.001 1.0001
f(x) 2.71 2.97 2.99 3 3.03 3.003 3.0012
∴ The function is
continuous at 1.

lim
𝑥→1
𝑥2
− 2𝑥 + 3
Yes
value at x = 1?
Yes

Checking of
Continuity
Is the function
defined at x =
1?
Yes
Does the limit
of the function
as x approache
s 1 exist?
Yes
Does the limit
of the function
as x approache
s 1 equal the
function value
at x = 1?
Yes
x 0.9 0.99 0.999 1 1.01 1.001 1.0001
f(x) 2.01 2.0001 2.000001 2 2.0001 2.000001 2.00000001
∴ The function is
continuous at 1.
lim
𝑥→1
𝑥2 − 2𝑥 + 3

lim
𝑥→1
𝑥3
− 𝑥
Yes
value at x = 1?
Yes

Checking of
Continuity
Is the function
defined at x =
1?
Yes
Does the limit
of the function
as x approache
s 1 exist?
Yes
Does the limit
of the function
as x approache
s 1 equal the
function value
at x = 1?
Yes
x 1.9 1.99 1.999 2 2.01 2.001 2.0001
f(x) 4.96 5.89 5.98 6 6.11 6.011 6.0011
∴ The function is
continuous at 2.
lim
𝑥→2
𝑥3 − 𝑥

DISCONTINUITY
TOPIC 9

D I S C O N T I N U I T Y
Discontinuity: a
point at which a
function is not
continuous

Discontinuity: a point at which a function is
not continuous

JUMP DISCONTINUITY
Jump Discontinuities: both one-sided
limits exist, but have different values.
The graph of f(x) below shows
a function that is discontinuous at x=a.
In this graph, you can easily see that
f(x)=L f(x)=M
The function is approaching different
values depending on the direction x is
coming from. When this happens, we say
the function has a jump
discontinuity at x=a.

● Functions with jump
discontinuities, when written out
mathematically, are
called piecewise
functions because they are
defined piece by piece.
● Piecewise functions are defined on
a sequence of intervals.
● You'll see how the open and
closed circles come into play with
these functions. Let's look at a
function now to see what a
piecewise function looks like.

𝑓 𝑥 = ቊ
𝑥 + 1, 𝑥 < −1
𝑥 + 4, 𝑥 ≥ −1
Left Side Limit
𝒂−
𝑥 + 1, 𝑥 < −1
Right Side Limit
𝒂+
𝑥 + 4, 𝑥 ≥ −1
−4 −3 −2 −1 0 1
−3 −2 −1 3 4 5

𝑓 𝑥 = ቊ
𝑥 + 1, 𝑥 < −1
𝑥 + 4, 𝑥 ≥ −1
Left Side Limit
𝒂−
𝑥 + 1, 𝑥 < −1
Right Side Limit
𝒂+
𝑥 + 4, 𝑥 ≥ −1
x −4 −3 −2 −1 0 1
f(x) −3 −2 −1 3 4 5

INFINITE DISCONTINUITY
The graph on the right shows a function that is
discontinuous at x=a.
The arrows on the function indicate it will grow
infinitely large as x approaches a. Since the
function doesn't approach a particular finite
value, the limit does not exist. This is an infinite
discontinuity.
The following two graphs are also examples of
infinite discontinuities at x=a. Notice that in all
three cases, both of the one-sided limits are
infinite.

REMOVABLE DISCONTINUITY
Two Types of Discontinuities
1) Removable (hole in the
graph)
2) Non-removable (break or
vertical asymptote)
A discontinuity is called removable if a
function can be made
continuous by defining (or
redefining) a point.

DIFFERENCE BETWEEN A
REMOVABLE AND NON REMOVABLE
DISCONTINUITY
If the limit does not exist, then
the discontinuity is non–
removable. In essence, if adjusting
the function’s value solely at the
point of discontinuity will render
the function continuous, then
the discontinuity is removable.

REMOVABLE DISCONTINUITY
Step 1: Factor the numerator and the
denominator.
Step 2: Identify factors that occur in
both the numerator and the
denominator.
Step 3: Set the common factors
equal to zero.
Step 4: Solve for x.

lim
𝑥→3
𝑓 𝑥 =
𝑥2
− 9
𝑥 − 3
denominator.
denominator.
equal to zero.
(𝑥 + 3)(𝑥 − 3)
𝑥 − 3
𝑥 + 3
𝑓 𝑥 = 3 + 3
lim
𝑥→3
𝑓 𝑥 =
𝑥2
− 9
𝑥 − 3
= 6

lim
𝑥→3
𝑓 𝑥 =
𝑥2
− 9
𝑥 − 3
Left Side Limit
𝒂−
𝑥 < 3
Right Side Limit
𝒂+
𝑥 > 3
2.9 2.99 2.999 3.01 3.001 3.0001
−5.9 −5.99 −5.999 6.01 6.001 6.0001

lim
𝑥→3
𝑓 𝑥 =
𝑥2
− 9
𝑥 − 3
Left Side
Limit
𝒂−
𝑥 < 3
Right Side
Limit
𝒂+
𝑥 > 3
2.9 2.99 2.999 3.01 3.0013.0001
5.9 5.99 5.999 6.01 6.001 6.0
001

lim
𝑥→2
𝑓 𝑥 =
𝑥2
− 5𝑥 + 4
𝑥2 − 4
denominator.
denominator.
equal to zero.
(𝑥 + 5)(𝑥 − 1)
(𝑥 + 2)(𝑥 − 2)
Non-removable, the limits DNE.

Continuity of a Composite Function
THEOREM CONTINUITY OF A COMPOSITE FUNCTION
If g is a continuous at a and f is a continuous at g(a) ,then the
composite function 𝑓°𝑔 𝑥 = 𝑓(𝑔 𝑥 ) is continuous at a.
Proof:
Because g is continuous at a,
𝑔(𝑥) = 𝑔(𝑎)
or, equivalently
Now f is continuous at g(a); thus, we apply Theorem to the composite 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏
𝑓°𝑔 𝑥 = 𝑓( 𝑔(𝑥))
= 𝑓(𝑔(𝑎)
= (𝑓°𝑔)(𝑎)

Illustrative Example:
Let f 𝑥 = 𝑥2
𝑎𝑛𝑑 𝑔 𝑥 = 𝑥 + 1
𝑓 𝑔 𝑥 = (𝑥 + 1)2
lim
𝑥→2
(𝑥 + 1)2
= (2 + 1)2
= 9
lim
𝑥→2
(𝑥 + 1)2
= 9
Illustrative Example:
Discuss the continuity of 𝑓(𝑔 𝑥 𝑤ℎ𝑒𝑟𝑒 f 𝑥 =
1
𝑥
𝑎𝑛𝑑 𝑔 𝑥 = 𝑥2
− 1
𝑓 𝑔 𝑥 = (𝑥 + 1)2
lim
𝑥→2
(𝑥 + 1)2
= (2 + 1)2
= 9
lim
𝑥→2
(𝑥 + 1)2
= 9

C O N T I N U I T Y
OF AN INTERVAL
TOPIC 10

CONTINUITY OF AN INTERVAL
A function is continuous on an interval if and only if we
can trace the graph of the function without lifting our pen
on the given interval

An interval is a set of real numbers
between two given numbers called the
endpoints of the interval
Finite Interval- intervals whose endpoints
are bounded
An open interval is one that does not
include its endpoints: 𝑎, 𝑏 𝑜𝑟 𝑎 < 𝑥 < 𝑏
A closed interval is one that includes its
endpoints
𝑎, 𝑏 𝑜𝑟 𝑎 ≤ 𝑥 ≤ 𝑏
Combination: composed of an open and
closed interval on either side: ሾ𝑎, 𝑏) 𝑜𝑟 𝑎 ≤
𝑥 < 𝑏; ( ሿ
𝑎, 𝑏 𝑜𝑟 𝑎 < 𝑥 ≤ 𝑏

Infinite Interval
These are intervals with at least one
unbounded side.
Open Left-bounded: 𝑎, ∞ 𝑜𝑟 𝑥 > 𝑎
Left side has an endpoint up to positive
infinity
Close Left bounded: ሾ𝑎, ∞) 𝑜𝑟 𝑥 ≥ 𝑎
Open Right Bounded: −∞, 𝑏 𝑜𝑟 𝑥 < 𝑏
Close Right Bounded: ( ሿ
−∞, 𝑏 𝑜𝑟 𝑥 ≤ 𝑏
Unbounded: −∞, ∞ 𝑜𝑟 𝑠𝑒𝑡 𝑜𝑓 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠

Interval
Notation
Set Notation
(2, 5) 2 < 𝑥 < 5
2, 5 2 ≤ 𝑥 ≤ 5
( ሿ
2,5 2 < 𝑥 ≤ 5
ሾ2,5) 2 ≤ 𝑥 < 5
−∞, 2 x<2
( ሿ
−∞, 2 x≤ 2
(5, ∞) x>5
ሾ5, ∞) 𝑥 ≥ 5
−∞, ∞ All Real Number

Continuity on an Open
Interval
A function is continuous on an
open interval if it is continuous for
any real number on that interval
Continuity on an
Closed Interval
A function f is continuous on a closed
interval 𝑎, 𝑏 if it is continuous on (a, b) and
lim
𝑥→𝑎=
𝑓 𝑥 = 𝑓 𝑎 𝑎𝑛𝑑 lim
𝑥→𝑏−
𝑓 𝑥 = 𝑓(𝑏)

Determine if the function is
continuous or not
1. 𝑓 𝑥 = 3𝑥 − 6, ( ሿ
−∞, 1
𝑥 ≤ 1
𝐶𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑜𝑛 𝑡ℎ𝑒 𝑔𝑖𝑣𝑒𝑛 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙

Determine if the function is
continuous or not
2. ℎ 𝑥
𝑥−5
𝑥2−𝑥−6
−1,4
−1 < 𝑥 < 4
Solve for x.
𝑥2 − 𝑥 − 6 = 0,
𝑥 − 3 𝑥 + 2 = 0
𝑥 = 3, 𝑥 = −2, x ≠ 3, 𝑥 ≠ −2
Discontinuous on the given interval

Determine if 𝑓 𝑥 = ቊ
2𝑥 − 1, 𝑥 < 5
𝑥2
, 𝑥 ≥ 5
is continuous on 2,5 𝑤ℎ𝑒𝑟𝑒 𝑎 =
2, 𝑏 = 5
Step 1: Check if continuous on (2,5)
We’ll us the subfunction that satisfy the
interval
Use f x = 2𝑥 − 1
Continuous for all real numbers

Continuity on an Closed Interval
A function f is continuous on a closed
interval 𝑎, 𝑏 if it is continuous on (a, b) and
lim
𝑥→𝑎+=
𝑓 𝑥 = 𝑓 𝑎 𝑎𝑛𝑑 lim
𝑥→𝑏−

2𝑥 − 1, 𝑥 < 5
𝑥2
, 𝑥 ≥ 5
2, 𝑏 = 5
Step 2: Check if lim
𝑥→𝑎+
𝑓 𝑥 = 𝑓(𝑎)
lim
𝑥→2+
𝑓 𝑥 = 2𝑥 − 1 = 2 2 − 1 = 3
Check 𝑓 2 = 2𝑥 − 1 = 2 2 − 1 = 3

2𝑥 − 1, 𝑥 < 5
𝑥2
, 𝑥 ≥ 5
2, 𝑏 = 5
𝑥→𝑏−
lim
𝑥→5−
𝑓 𝑥 = 2𝑥 − 1 = 2 5 − 1 = 9
Check 𝑓 5 = 𝑥2 = 52 = 25
Discontinuous on
the interval 2,5

Determine if 𝑓 𝑥 = ൝
𝑥 − 6, 𝑥 < 2
𝑥+1
𝑥−1
𝑥 ≥ 2
is continuous on −3,1 𝑤ℎ𝑒𝑟𝑒 𝑎 = −3 𝑏 = 1
Step 1: Check if continuous on (-3,1)
interval
Use f x = 𝑥 − 6

𝑥 − 6, 𝑥 < 2
𝑥+1
𝑥−1
𝑥 ≥ 2
𝑥→𝑎+
lim
𝑥→−3+
𝑓 𝑥 = 𝑥 − 6 = −3 − 6 = 9
Check 𝑓 −3 = 𝑥 − 6 = −3 − 6 = 9

𝑥→𝑏−
lim
𝑥→1−
𝑓 𝑥 = 𝑥 − 6 = 1 − 6 = −5
Check 𝑓 1 = 𝑥 − 6 = 1 − 6 = −5
Continuous on the
interval −3,1
𝑥 − 6, 𝑥 < 2
𝑥+1
𝑥−1
𝑥 ≥ 2

Determine if 𝑓 𝑥 = ቊ𝑥2
− 9, 𝑥 ≥ 4
𝑥 + 5 < 4
is continuous on 1,4 𝑤ℎ𝑒𝑟𝑒 𝑎 = 1 𝑏 = 4
Step 1: Check if continuous on (1,4)
interval
Use f x = 𝑥 + 5

− 9, 𝑥 ≥ 4
𝑥 + 5 < 4
𝑥→𝑎+
lim
𝑥→1+
𝑓 𝑥 = 𝑥 + 5 = 1 + 5 = 6
Check 𝑓 1 = 𝑥 + 5 = 1 + 5 = 6

− 9, 𝑥 ≥ 4
𝑥 + 5 < 4
𝑥→𝑏−
lim
𝑥→4−
𝑓 𝑥 = 𝑥 + 5 = 4 + 9 = 9
Check 𝑓 4 = 𝑥2
− 9 = 42
− 9 = 16 − 9 = 7
Discontinuous on
the interval 1,4

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