1.4
Continuity
and
One-Sided Limits
This
will test
the
“Limits”
of your
brain!
Definition of Continuity
A function is called continuous at c if the
following three conditions are met:
1.
f(c) is defined
2.
lim f ( x) exists
3.
lim f ( x)  f (c)
x c
x c
A function is continuous on an open
interval (a,b) if it is continuous at each
point in the interval.
Two Types of Discontinuities
1. Removable
Point Discontinuity
2. Non-removable
Jump and Infinite
Removable example
x2 1
f ( x) 
x 1
( x  1)( x  1)
f ( x) 
x 1
the open circle
can be filled in
to make it
continuous
Non-removable discontinuity.
Ex.
lim
x
 -1
lim
x 0
lim
x 0
x
x
x
x 0
 1
x
x
Determine whether the following functions are
continuous on the given interval.
1
f ( x)  , 0,1
x
yes, it is
continuous
(
)
1
x 1
f ( x) 
, (0,2)
x 1
2
(
)
discontinuous at x = 1
removable discontinuity since filling in (1,2)
would make it continuous.
f ( x)  sin x, (0,2 )
2
yes, it is continuous
One-sided Limits
lim f ( x)  L
Limit from the right
lim f ( x)  L
Limit from the left
x c
x c
Find the following limits
lim x  1
0
lim x  1
D.N.E.
x 1
x 1
lim
x 1
x 1
1
D.N.E.
Left  Right
Step Functions
Greatest Integer
“Jump”
f ( x)  x
lim x  
-1
lim x  
0
x 0
x 0
lim x  
x 0
D.N.E.
Left  Right
5  x , 1  x  2
g(x)=
x  1,
2
2 x3
Is g(x) continuous at x = 2?

5  x 
lim g ( x)  xlim

2
3
x2

x  1 
lim g ( x)  xlim
2
2
3
x2
 g(x) is continuous at x = 2
Intermediate Value Theorem
If f is continuous on [a,b] and k is any number
between f(a) and f(b), then there is at least one
number c in [a,b] such that f(c) = k
f(a)
In this case, how
many c’s are there
where f(c) = k?
k
f(b)
[
a
]
b
3
Show that f(x) = x3 + 2x –1
has a zero on [0,1].
f(0) = 03 + 2(0) – 1 = -1
f(1) = 13 + 2(1) – 1 = 2
Since f(0) < 0 and f(1) > 0, there must
be a zero (x-intercept) between [0,1].