2.3
Continuity
Quick Review
3x  2 x  1
1. Find lim
x 1
x3  4
2. Let f x  int x. Find each limit.
a. lim  f x   2 b. lim  f x 
2
2
x  1
c. lim f x 
x 1
x  1
DNE
d . f  1
 x 2  4 x  5, x  2
3. Let f x   
4  x, x  2

Find each limit.
a. lim f x 
b. lim f x 
1
x 2
c. lim f x 
x 2
x 2
DNE
d . f 2
1
1
2
2
Quick Review
In Exercises 4 – 6, find the remaining functions in the list of functions:
f , g, f o g, g o f.
2x  1
1
4. f x  
, g x    1
x5
x
x2
3x  4
 f  g x  
, x  0 g  f x  
, x  5
6x  1
2x  1
5. f x   x 2 , g  f x   sin x 2 , domain of g  0, 
g x   sin x, x  0
f
 g x   sin 2 x, x  0
Quick Review
In Exercises 4 – 6, find the remaining functions in the list of functions:
f , g, f o g, g o f.
1
6. g x   x  1, g  f x   , x  0
x
1
x
f x   2  1, x  0
 f  g x  
, x 1
x
x 1
7. Use factoring to solve 2 x 2  9 x  5  0.
1
x  , 5
2
8. Use graphing to solve x3  2 x  1  0.
x  0.453
Quick Review
In Exercises 9 and 10, let

5  x, x  3
f x    2
  x  6 x  8, x  3
9. Solve the equation f x  4.
x 1
10. Find a value of c for which the equation f x  c
has no solution.
Any c in 1, 2
What you’ll learn about





Continuity at a Point
Continuous Functions
Algebraic Combinations
Composites
Intermediate Value Theorem for Continuous
Functions
Essential Question
How can continuous functions be used to describe
how a body moves through space and how the
speed of a chemical reaction changes with time?
Continuity at a Point
Any function y  f  x  whose graph can be sketched in one continuous motion
without lifting the pencil is an example of a continuous function.
Example Continuity at a Point
1. Find the points at which the given function is continuous and the
points at which it is discontinuous.
The function is continuous at 0 < x < 2,
and 3 < x < 6.
The function is discontinuous at x < 0,
2 < x < 3, and x > 6.
Continuity at a Point
Interior Point: A function y  f  x  is continuous at an interior point c of its
domain if lim f  x   f  c 
x c
Endpoint: A function y  f  x  is continuous at a left
endpoint a or is continuous at a right endpoint b of its domain if
lim f  x   f  a 
x a
or
lim f  x   f  b 
x b 
respectively.
If a function f is not continuous at a point c , we say that f is
discontinuous at c and c is a point of discontinuity of f.
Note that c need not be in the domain of f.
Continuity at a Point
The typical discontinuity types are:
Removable
Jump
Infinite
Oscillating
Example Continuity at a Point
2. Given the graph of f (x), shown below, determine if f (x) is continuous
at x = 2, x = 0, x = 3. Then name the type of discontinuity at each
point.
Discontinuous at x = – 2
Jump Disc.
Continuous at x = 0
Discontinuous at x = 3
Removable Disc.
Example Continuity at a Point
3. Write a piecewise function for the given function to remove any point
of removable discontinuity.
x  33x  2x  1
x  7x  6
f x  
2
x 9
x  3 x  3

x  2x  1 20 10


lim
x 3
6
x  3
3
 x3  7 x  6
, x3

2
x

9
f x   
10

, x3

3
Continuous Functions
A function is continuous on an interval if and only if it is
continuous at every point of the interval. A continuous
function is one that is continuous at every point of its domain.
A continuous function need not be continuous on every interval.
4. Is the following function a continuous function?
3
f x  
x 1
The function is a continuous function because it is
continuous at every point of its domain.
It does have a point of discontinuity at x = – 1
because it is not defined there.
Properties of Continuous Functions
If the functions f and g are continuous at x  c, then the
following combinations are continuous at x  c.
1. Sums :
f g
2. Differences:
f g
3. Products:
f g
4. Constant multiples: k  f , for any number k
f
5. Quotients:
, provided g  c   0
g
Composite of Continuous Functions
If f is continuous at c and g is continuous at f  c  , then the
composite g f is continuous at c.
5. Use the Theorem to show that the given function is continuous?


f x   sin x  1
2
Let g x   sin x and
f x   x  1
2
Since we know the sin function is continuous and
the polynomial f (x) is continuous, the composition
function is continuous.
Intermediate Value Theorem
Suppose that f (x ) is continuous on [a, b] and let M be any
number between f (a ) and f (b ). Then there exists a number c
such that,
1. a  c  b
2. f c   M
The Intermediate Value Theorem tells us that a function will
take the value of M somewhere between a and b but it doesn’t
tell us where it will take the value nor does it tell us how many
times it will take the value. These are important ideas to
remember about the Intermediate Value Theorem.
Intermediate Value Theorem
6. Show that the following function has a root somewhere in the
interval [-1, 2].
px   2 x3  5x 2  10 x  5
p 1  2 1  5 1  10 1  5  8
3
2
p2  22  52  102  5  19
3
2
Since we know polynomial p (x) is continuous, by the
Intermediate Value Theorem, there must be values of x in
the given interval that have outputs between –19 and 8.
Therefore there is a value of x in the interval that will
have an output of 0.
Intermediate Value Theorem
7. If possible, determine if the following function takes the
following values in the interval [0, 5].
 x2 
f x   20 sin x  3 cos 
 2
(a) Does f (x) = 10?
(b) Does f (x) = –10?
yes
Cannot determine
 02 
f 0  20 sin 0  3 cos   2.822
2
 52 
f 5  20 sin 5  3 cos   19.744
2
Pg. 84, 2.3 #2-50 even