1.
2.
3.
4.
Objectives:
Be able to define continuity by determining if a graph is
continuous.
Be able to identify and find the different types of
discontinuities that functions may contain.
Be able to determine if a function is continuous on a closed
interval.
Be able to determine one-sided limits and continuity on a
closed interval.
Critical Vocabulary:
Limit, Continuous, Continuity, Composite Function
1. Page 237 #23-43 odd, 49-55 odd, 61, 63, 77
2. Page 236 #1-17 odd, 79, 88
I. Continuity
Continuous: To say that a function f is continuous at x = c
there is no interruption in the graph of f at c
This means a graph will contain no HOLES, JUMPS, or GAPS
Simple Terms:
If you ever have to lift your pencil to sketch a graph, then it is
not continuous.
I. Continuity
What Causes discontinuity?
1. The function is not defined at c.
Let’s look at at f(x) = ½x - 2
This is an example of a
hole in the graph at f(-2)
Concept: The function is
not defined at c.
f(c) = not defined
c
I. Continuity
What Causes discontinuity?
2. The limit of f(x) does not exist at x = c
Let’s look at at
1
 x 1 x  3
f ( x)   3
 x  7 x  3
This is an example of
a gap in the graph at
x=3
c
Concept: The limit does
not exist at x = c
I. Continuity
What Causes discontinuity?
3. The limit of f(x) exists at x = c but is not equal to f(c).
lim f ( x)  f (c)
x c
Let’s look at the first graph again
This is an example of
a jump in the graph
What is the limit as x
approaches -2?
What is f(-2)?
c
Concept: The behavior
(limit) and where its
defined (f(c)) are not
the same.
I. Continuity
Continuous: To say that a function f is continuous at x = c
there is no interruption in the graph of f at c
This means a graph will contain no HOLES, JUMPS, or GAPS
Simple Terms:
If you ever have to lift your pencil to sketch a graph, then it is
not continuous.
A function f is continuous at c if the following three conditions
are met:
1. f(c) is defined
2.
lim f ( x ) exists
x c
3.
lim f ( x)  f (c)
x c
1.
2.
3.
4.
Objectives:
Be able to define continuity by determine if a graph is
continuous.
Be able to identify and find the different types of
discontinuities that functions may contain.
Be able to determine if a function is continuous on a closed
interval.
Be able to determine one-sided limits and continuity on a
closed interval.
Critical Vocabulary:
Limit, Continuous, Continuity, Composite Function
II. Discontinuities
When you are asked to “discuss the continuity” of each
function, you are really being asked to describe any place
where the graph is discontinuous.
Discontinuity is broken into 2 Categories:
1. Removable: A discontinuity is removable if you
COULD define f(c).
c
c
HOLES
JUMPS
II. Discontinuities
When you are asked to “discuss the continuity” of each
function, you are really being asked to describe any place
where the graph is discontinuous.
Discontinuity is broken into 2 Categories:
1. Removable: A discontinuity is removable if you
COULD define f(c).
2. Non-Removable: A discontinuity is non-removable if you
CANNOT define f(c).
c
c
GAPS
ASYMPTOTES
II. Discontinuities
When you are asked to “discuss the continuity” of each
function, you are really being asked to describe any place
where the graph is discontinuous.
Discontinuity is broken into 2 Categories:
1. Removable: A discontinuity is removable if you
COULD define f(c).
2. Non-Removable: A discontinuity is non-removable if you
CANNOT define f(c).
x2 1
Example 1: f ( x) 
What is the Domain?  ,11, 
x 1
f ( x) 
( x  1)( x  1)
x 1
Linear Function
Has a Removable discontinuity at x = -1
Specific: Hole at (-1, -2)
What intervals is the graph continuous?
 ,11, 
II. Discontinuities
When you are asked to “discuss the continuity” of each
function, you are really being asked to describe any place
where the graph is discontinuous.
Discontinuity is broken into 2 Categories:
1. Removable: A discontinuity is removable if you
COULD define f(c).
2. Non-Removable: A discontinuity is non-removable if you
CANNOT define f(c).
x3
What is the Domain?
x2  9
 ,3(3,3)3, 
x 3
f ( x) 
Rational Function
( x  3)( x  3)
Example 2: f ( x) 
Has a Removable discontinuity at x = 3
Specific: Hole at (3, 1/6)
Has a Non-Removable discontinuity at x = -3
What intervals is the graph continuous?
 ,3(3,3)3, 
II. Discontinuities
When you are asked to “discuss the continuity” of each
function, you are really being asked to describe any place
where the graph is discontinuous.
Discontinuity is broken into 2 Categories:
1. Removable: A discontinuity is removable if you
COULD define f(c).
2. Non-Removable: A discontinuity is non-removable if you
CANNOT define f(c).
Example 3: Discuss the continuity of the composite
function f(g(x))
1
g ( x)  x  1
x
1
f ( g ( x))  f ( x  1) 
x 1
1
f ( g ( x)) 
x 1
f ( x) 
x+1>0
x > -1
What intervals is the graph continuous?
1, 
II. Discontinuities
Example 5: Graph the piecewise function, then determine on
which intervals the graph is continuous.
 x 1 x  0
f ( x)   2
x  2 x  0
Non-Removable discontinuity at x = 0
What intervals is the graph continuous? ,00, 
III. Closed Intervals
Closed Interval: Focusing on specific portion (domian) of a graph.
[a, b]
Example 5: Discuss the continuity on the closed interval.
f ( x) 
1
x2  4
 1,2
Non-Removable discontinuity at x = 2
What intervals is the graph continuous?  1,2
1. Page 237 #23-43 odd, 49-55 odd, 61, 63, 77
2. Page 236 #1-17 odd, 79, 88
1.
2.
3.
4.
Objectives:
Be able to define continuity by determine if a graph is
continuous.
Be able to identify and find the different types of
discontinuities that functions may contain.
Be able to determine if a function is continuous on a closed
interval.
Be able to determine one-sided limits and continuity on a
closed interval.
Critical Vocabulary:
Limit, Continuous, Continuity, Composite Function
IV. One-Sided Limits
What does a One-Sided look like?
lim f ( x)  L
c
lim f ( x)  L
c
Approach from the
right only
lim f ( x)  L
c
Approach from the
left only
x c
x c
x c
Example 1: Graph
f ( x)  4  x 2
IV. One-Sided Limits
Example 1: Graph
f ( x)  4  x 2
then find the limits
What’s the domain?
x
-2
f(x)
0
-1
3
0
2
1
2
3 0
lim f ( x)  ____
DNE
x2
0
lim f ( x)  ____
x2
2
lim f ( x)  ____
x 0
0
lim  f ( x)  ____
DNE
lim f ( x)  ____
lim  f ( x)  DNE
____
lim f ( x)  DNE
____
x  2
x  2
x2
x  2
IV. One-Sided Limits
 4 x
Example 1: Graph f ( x)  
2
4
x

x

x
1
2
3
4
f(x) 3 4
3
0
x 1
x 1
then find the limits
3
lim f ( x)  ____
x 1
3
lim f ( x)  ____
x 1
lim f ( x)  ____
3
x 1
Is this graph continuous?
Has a Removable discontinuity at x = 3
Specific: Hole at (1, 3)
1. Page 237 #23-43 odd, 49-55 odd, 61, 63, 77
2. Page 236 #1-17 odd, 79, 88