Name: __________________________________ Period _______ Sec1.4
Continuity at a point and on open or closed intervals
Assignment: Sec1.4 Page 78-79 #1-21 odd, 27-30 all, 31-51 odd
Objective: In this lesson you will learn how to determine continuity at a point, on an open interval,
and on a closed interval. You will also learn how to determine one-sided limits.
Recall the difference between open and closed intervals. How do you write these intervals using
inequalities and interval notation?
Open Interval
Closed Interval
All real numbers
Inequality notation:
Inequality notation:
Inequality notation:
Interval Notation:
Interval notation:
Interval notation:
Important Concept: Continuity
If a function is continuous at x = c that means that there is no interruption in the graph at x = c. We
often think about this as if we can sketch the graph without lifting the pencil off the paper.
That is, the graph has no holes, jumps or gaps. Look at the three examples to see how continuity can
be destroyed by any of these conditions.
(1) There is a HOLE
The function is not defined at x=c
(2) There is a JUMP:
(Infinite jumps happen too)
The limits does not exists at x = c
(3) There is a GAP
The limit does exists at x=c, but the
limit doesn’t equal the y-value at x=c.
If NONE of these conditions happens at x = c, we say that the function is continuous at the point x=c.
Definition of Continuity at a point, x=c
(1) f (c ) is defined
This means _______________________________________________.
(2)
(3)
lim
xc
lim
xc
f ( x)
exists
f ( x )  f (c )
This means _______________________________________________.
This means _______________________________________________.
Definition of Continuity on an open interval (a,b):
A function is continuous on an open interval (a,b) if _________________________________________
__________________________________________________________________________________.
If a function is continuous everywhere that means __________________________________________.
Examples: Discuss the continuity of each function
1
x2 1
(a) f ( x) 
(b) g ( x) 
x 1
x 1
Domain:
Domain:
 x  1, x  0
(c) h( x)   2
x 1 x  0
Domain:
(d) t ( x)  cos x
Sketch the graph:
Sketch the graph
Sketch the graph
Sketch the graph
Continuous?
Continuous?
Continuous?
Continuous?
If not, which reason
destroys the
continuity?
If not, which reason
destroys the continuity?
If not, which reason
destroys the continuity?
If not, which
reason destroys
the continuity?
Domain:
Before you can understand continuity on a closed interval, you need to understand what a one-sided
limit means and be able to evaluate one-sided limits.
Limit from the right:
Limit from the left:
We say that the limit exists,
lim
f ( x) 
x  c
lim
x  c
lim
f ( x) 
f ( x) 
xc
if
lim
x  c
f ( x)
=
lim
x  c
f ( x)
Example:
lim
Limit from the right:
f ( x) 
x  c
lim
Limit from the left:
x  c
f ( x) 
lim
Then, the limit is
f ( x) 
xc
Example: Find the limit (if it exists). If it does not exist, explain why.
Look at the graph of y  x for x-values close to x = 0.
(a)
x
lim
x0
(b)

x
lim
x0

(c)
x
lim
x0
Example (like #12): Find the limit (if it exists). If it does not exist, explain why.
x 3
Look at the graph of y 
for x-values close to x = 9.
x9
(a)
x 3

x 9
lim
x9
(b)

x 3

x9
lim
x9
(c)

lim
x 3

x 9
x9
Definition of Continuity on an closed interval [a,b]
A function is continuous on a closed interval if
(1) It is continuous on a open interval (a,b), and
(2) The function is continuous from the right at x = a, which means
lim
f ( x)  f ( a )
x  a
lim
f ( x)  f (b)
(3) The function is continuous from the left at x = b, which means

xb
Example: Greatest Integer function, f ( x)  x
Find these limits:
lim
x2
x 

The y-value is the greatest integer n such that n  x
x 
lim
x2

Does the limit exists as x approaches 2? Why or why not?
Is the function continuous on the open interval, (1,2)? ______
Is the function continuous on the closed interval [1,2]? ______
Is the function continuous on the interval [1,2)? _______
Is the function continuous on the interval (1,2]? _______
Testing for continuity (Recall that continuity is destroyed if you have hole, jump or gap!)
Hole (these are also called removable discontinuities): No y-value at x
Jump: There is no limit
Gap: There is a limit and a y-value, but they are not equal
Strategy:
Look at the graph. If you have a piece-wise function, look at the y-values at the transition point.
Example #51: Find the x-values (if any) at which f is not continuous.
Which of the discontinuities are removable?
 x, x  1
f ( x)   2
x , x  1
Work some additional examples:
Possible limits: #16 or #18
Possible continuity problems: #36, 42, 52