Continuity and
One-Sided Limits
Lesson 3.4
Continuity
Continuity – A function f is continuous at c if…
f ( x ) exists 3) lim f ( x)  f (c)
1) f (c) is defined 2) lim
x c
x c
If every point along the interval (a, b) is continuous,
then f is continuous along the interval.
f (c) is undefined
a
c
b
limit does not
exist at c
a
c
b
lim f ( x)  f (c)
x c
a
c
b
Examples of Continuity
Describe each function’s continuity.
1) f ( x)  x 2
-1
1
f(x) is continuous
across its domain
2) g ( x)  x 2 / x
-1
1
g(x) has removable
discontinuity at
x=0
3) h( x)  1 / x
-1
1
 x  1, x  0
2
 x  1, x  0
4) k ( x)  
-1
1
h(x) has
k(x) is continuous
nonremovable
across (-∞,0) and (0,∞).
discontinuity at x = 0
k ( x)  1 ,
Because lim
x 0
the piecewise function
k(x) is continuous
across its domain.
One-Sided Limits
One-Sided Limits – a value defined as f (x) becomes
arbitrarily close to a single number L as x approaches c from
values either larger or smaller than x
lim f ( x ) does not exist, but…
x c
y
lim f ( x)  y
x c
z
a
As x  c from
the left…
c
b
As x  c from
the right…
lim f ( x)  z
x c
Note: lim f ( x )  z only exists IF…
x c
lim f ( x)  lim f ( x)  L
x c
x c
One-Sided Limit Examples
Evaluate each expression.
f ( x) = 1
1) xlim
1
-1
1
As x  1 from
the right…
g ( x) = -1
2) xlim
0 
-1
1
h( x) = -2
3) xlim
 1
As x  0 from As x-1 from
the right…
the left…
-1
1
Open and Closed Intervals
closed interval – a continuous range of
values containing its endpoints, ex:
-2 ≤ x ≤ 2, otherwise denoted as [-2, 2]
-2
2
-2
2
open interval – a continuous range of
values excluding its endpoints, ex:
-2 < x < 2, otherwise denoted as (-2, 2)
Intermediate Value Theorem
Intermediate Value Theorem – If f is continuous on the
closed interval [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.
1) If f is continuous across [a, b]…
2) f (a) < k < f (b)…
f(b)
k
 There MUST be a c, a < c < b, so
that f (c) = k
f(a)
a
c
b
Lesson Practice
Find each limit.
1a)
lim 16  x
x 4
2
1b) lim 16  x 2
x 4
2) Describe the continuity of the function
f ( x) 
x
x2  x
Classwork
Pg. 234 (3-11 odd; 17, 19, 23, 24, 37, 44, 74, 76)