Continuity

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Continuity and One-Sided Limits
Lesson 2.4
2
3
4
Intuitive Look at Continuity
 A function without
breaks or
jumps

 The graph can be
drawn without lifting the pencil
5
Continuity at a Point
 A function can be discontinuous at a point
• A hole in the function and the function not
defined at that point
• A hole in the function, but the function is
defined at that point
6
Continuity at a Point
 A function can be discontinuous at a point
• The function jumps to a different value at a
point
• The function goes to infinity at one or both
sides of the point, known as a pole
7
Definition of Continuity at a Point

A function is continuous at a point x = c
if the following three conditions are met
1. f(c) is defined
f ( x) exists
2. lim
x c
f ( x )  f (c )
3. lim
x c

x=c
For this link, determine which of the
conditions is violated in the examples of
discontinuity
8
"Removing" the Discontinuity
A discontinuity at c is called removable if …
 If the function can be made continuous by
• defining the function at x = c
• or … redefining the function at x = c
 Go back to this link, determine which (if
any) of the discontinuities can be removed
9
Which of These is Dis/Continuous?
 When x = 1 … why or not
x  2x  3
f ( x) 
x 1
2
x2  2 x  3
g ( x) 
x 1
if x ≠ 1 and
g(x) = 6 if x = 1
Are any removable?
x2  2 x  3
h( x ) 
x 1
if x ≠ 1 and
h(x) = 4 if x = 1
x3
F ( x) 
x 1
if x ≠ 1 and
F(x) = 4 if x = 1
10
Continuity Theorem
 A function will be continuous at any
number
x = c for which f(c) is defined, when …





f(x) is a polynomial
f(x) is a power function
f(x) is a rational function
f(x) is a trigonometric function
f(x) is an inverse trigonometric function
11
Properties of Continuous Functions
 If f and g are functions, continuous at x = c
Then …
 s  f ( x ) is continuous (where s is a constant)


f(x) + g(x) is continuous
f ( x)  g ( x) is continuous

f ( x)
is continuous
g ( x)

f(g(x)) is continuous
12
One Sided Continuity
 A function is continuous from the right at a
point x = a if and only if lim f ( x)  f (a)
x a
 A function is continuous from the left at a
point x = b if and only if lim f ( x)  f (b)
x b
a
b
13
Continuity on an Interval
 The function f is said to be continuous on
an open interval (a, b) if
• It is continuous at each number/point of the
interval
 It is said to be continuous on a closed
interval [a, b] if
• It is continuous at each number/point of the
interval and
• It is continuous from the right at a and
continuous from the left at b
14
Continuity on an Interval
 On what intervals are the following
functions continuous?
g ( x)  sin x
x2 1
f ( x)  2
x 4
15
Intermediate Value Theorem
 Given function f(x)
• Continuous on closed interval [a, b]
• And L is a number strictly between f(a) and f(b)
 Then … there exists at least one number c
on the open interval (a, b) such that f(c) = L
f(b)
L
f(a)
a
c
b
16
Locating Roots with Intermediate
Value Theorem
 Given f(a) and f(b) have opposite sign
• One negative, the other positive
 Then there must be a root between a and b
a
b
Try exercises 88,
90, and 92 : pg 100
17
Assignment
 Lesson 2.4A
 Page 98
 Exercises 1 – 49 odd
 Lesson 2.4B
 Page 99
 Exercises 51 – 75, 85 – 105 odd
18
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