Continuity
Objective: To define what makes a
function continuous
Definition of Continuity
• A graph will have a break or a hole if any of the
following conditions occur.
1. The function is undefined at a point (hole or
asymptote).
Definition of Continuity
• A graph will have a break or a hole if any of the
following conditions occur.
1. The function is undefined at a point (hole or
asymptote).
2. The two sided limit at a value does not exist.
Definition of Continuity
• A graph will have a break or a hole if any of the
following conditions occur.
1. The function is undefined at a point (hole or
asymptote).
2. The two sided limit at a value does not exist.
3. The value at the point and the two-sided limit are
different.
Definition of Continuity
• A function is said to be continuous at x = c provided
the following conditions are satisfied:
1. f (c ) is defined.
f ( x ) exists.
2. lim
x c
f ( x )  f (c )
3. lim
x c
Definition of Continuity
• A function is said to be continuous at x = c provided
the following conditions are satisfied:
1. f (c ) is defined.
f ( x ) exists.
2. lim
x c
f ( x)  f (c )
3. lim
x c
In other words, the two-sided limit must exist
and be equal to the value at c.
Definition of Continuity
• A function is said to be continuous at x = c provided
the following conditions are satisfied:
1. f (c ) is defined.
f ( x ) exists.
2. lim
x c
f ( x)  f (c )
3. lim
x c
We only need to check condition 3. If condition 3
fails to hold, we say that f has a
discontinuity at x = c.
Example 1a
• Determine whether the following functions are
continuous at x = 2.
x2  4
f ( x) 
x2
Example 1a
• Determine whether the following functions are
continuous at x = 2.
x2  4
f ( x) 
x2
f (2) 
0
 hole
0
• Not continuous. Not defined at x = 2.
Example 1b
• Determine whether the following functions are
continuous at x = 2.
x2  4
,x  2
g ( x)  x  2
x2
3,
Example 1b
• Determine whether the following functions are
continuous at x = 2.
x2  4
,x  2
g ( x)  x  2
x2
3,
• Not continuous. Value and two sided limit are
different.
Example 1c
• Determine whether the following functions are
continuous at x = 2.
x 4
,x  2
g ( x)  x  2
x2
4,
2
Example 1c
• Determine whether the following functions are
continuous at x = 2.
x 4
,x  2
g ( x)  x  2
x2
4,
2
• Continuous!
Example 1
• The discontinuities in this example are removable
discontinuities.
• In example 1b, we can redefine the value of f(2) = 3
to f(2) = 4. This removes the discontinuity.
• In example 1c, we defined the value of f(2) = 4
Removable Discontinuity
• Where a graph has an asymptote, it will always be
discontinuous.
• Where a graph has a hole, we say that it has a
removable discontinuity.
Continuity on an Interval
• If a function is continuous at each number in an open
interval (a, b), then we say it is continuous on (a, b).
Continuity on an Interval
• If a function is continuous at each number in an open
interval (a, b), then we say it is continuous on (a, b).
• We can look at an open interval and say that a
function is continuous from the right or continuous
from the left.
• To be continuous from the right/left, the one-sided
limit and the value must be the same.
Continuity on an Interval
• If a function is continuous at each number in an open
interval (a, b), then we say it is continuous on (a, b).
• We can look at an open interval and say that a
function is continuous from the right or continuous
from the left.
• To be continuous from the right/left, the one-sided
limit and the value must be the same.
• The graph is continuous from the left
• Not continuous from the right.
Continuity on an Interval
• Definition 1.5.2
• A function f is said to be continuous on a closed
interval [a, b] if the following conditions are satisfied:
1. f is continuous on (a, b)
2. f is continuous from the right at a.
3. f is continuous from the left at b.
Example 2
• What can we say about the continuity of the
function f ( x)  9  x 2 ?
Example 2
• What can we say about the continuity of the
function f ( x)  9  x 2 ?
• The natural domain of this function is [-3, 3], so we
need to check (-3, 3) and both endpoints.
Example 2
• What can we say about the continuity of the
function f ( x)  9  x 2 ?
• The natural domain of this function is [-3, 3], so we
need to check (-3, 3) and both endpoints. Since
f (3)  lim 9  x 2  0 and f (3)  lim 9  x 2  0
x3
the function is continuous on [-3, 3].
x 3
Properties of Continuous Functions
• Theorem 1.5.3 If the functions f and g are continuous
at c, then
(a) f + g is continuous at c.
(b) f – g is continuous at c.
(c) fg is continuous at c.
(d) f/g is continuous at c if g(c) is not zero and has a
discontinuity at c if g(c) = 0.
Continuity of Polynomials/Rational
Functions
• Theorem 1.5.4
(a) A polynomial is continuous everywhere.
(b) A rational function is continuous at every point
where the denominator is nonzero, and has
discontinuities where the denominator is zero.
Example 3
• For what values of x is there a discontinuity in the
x2  9
graph of
x  5x  6
2
?
Example 3
• For what values of x is there a discontinuity in the
x2  9
graph of
x  5x  6
2
Since
?
x2  9
( x  3)( x  3)

2
x  5 x  6 ( x  3)( x  2)
we have a removable discontinuity at x = 3, and a
non-removable discontinuity at x = 2.
Continuity of Compositions
• Theorem 1.5.5
g ( x)  L and if the function f is continuous at L,
• If lim
x c
f ( g ( x))  f ( L) . That is
then lim
x c

lim f ( g ( x))  f lim g ( x)
x c
x c

Continuity of Compositions
• Theorem 1.5.6
(a) If the function g is continuous at c, and the function
f is continuous at g(c), then the composition f  g
is continuous at c.
(b) If the function g is continuous everywhere and the
function f is continuous everywhere, then the
composition f  g is continuous everywhere.
The Intermediate-Value Theorem
• Theorem 1.5.7
• If f is continuous on a closed interval [a, b] and k is
any number between f(a) and f(b), inclusive, then
there is at least one number x in the interval [a, b]
such that f(x) = k.
Homework
• Pages 118-119, 1-31 odd