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ContinuitySection 2.3
Review Quiz
Review Quiz (Continued)
2 2
In Exercises 4 – 6, find the remaining functions in the list of functions:
, , , .
2 1 14. , 1
5
5. , sin , domain of [0, )
f g f g g f
xf x g x
x x
f x x g f x x g
2
3
16. 1, , 0
7. Use factoring to solve 2 9 5 0
8. Use graphing to solve 2 1 0
g x x g f x xx
x x
x x
Review Quiz (Answers)
2
31
1 1
1
2
2 2
2
2
2 1
no limi
3 2 11. Find lim
4
2. Let int . Find each limit.
a lim b lim
c lim d 1
4 5, 23. Let
4 , 2
Find each limit.
a lim b lim
l
t
1 2
i
1
c m
x
x x
x
x x
x
x x
x
f x x
f x f x
f x f
x x xf x
x x
f x f x
dno limit 22 f x f
Review Quiz (Answers)
2 2
2
In Exercises 4 – 6, find the remaining functions in the list of functions:
, , , .
2 1 14. , 1
5
5. , sin , domain of [0, )
2 3 4, 0 , 5
6 1 2 1
sin , 0 sin , 0
f g f g g f
xf
x xf g x x
x g xx x
f x
g f x xx x
g x x x f g x x x
x g f x x g
Review Quiz (Answers)
2
2
3
16. 1, , 0
7. Use factoring to solve 2 9 5 0
8. Use graphing to solve 2 1 0
11, 0 , 1
1
1, 5
2
0.453
xf x x f g x
g x x g f x xx
x x
x x
xxx
x
x
What you’ll learn . . .• Continuity at a point
• Continuous Function
• Algebraic Combinations
• Composites
• Intermediate Value Theorem for Continuous Functions
Continuity at a Point
Recall, 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. You have seen this in both Algebra II and Pre-Calculus.
Interior Point: A function is continuous at an interior point of its
domain if lim
Endpoint: A function is continuous at a left
endpoint or is continuous
x c
y f x c
f x f c
y f x
a
at a right endpoint of its domain if
lim or lim respectively.x a x b
b
f x f a f x f b
Definition: Continuity
A function is continuous at x = a when (i) f(a) is defined(ii) lim𝑥→𝑎 𝑓 𝑥 exists(iii) lim
𝑥→𝑎𝑓 𝑥 = f(a)
Otherwise, f is said to be discontinuous at x = a.
Types of Discontinuity
• REMOVABLE
• JUMP
• INFINITE
• OSCILLATING
Example: Continuity at a Point
2
3Find and identify the points of discontinuity of
1y
x
There is an infinite discontinuity at 1.x
Continuous Function
A function is if and only if
it is continuous at every point of the interval. A
is one that is continuous at every
point of its domain. A continuous funct
continuous on an interval
continuous function
ion need not be
continuous on every interval.
Example: Finding where a Rational Function is Continuous
Determine where 𝑓 𝑥 =𝑥2+2𝑥−3
𝑥−1is continuous
Example: Removing a DiscontinuityMake the function from the last slide continuous everywhere by redefining it at a single point.
Example: Nonremovable Discontinuity
Find all discontinuities of 𝑓 𝑥 =1
𝑥2and 𝑔 𝑥 = cos
1
𝑥.
Consider . . . All polynomials are continuous everywhere. Additionally, sinx, cos x, 𝑡𝑎𝑛−1(𝑥) and 𝑒𝑥are continuous everywhere. 𝑛 𝑥 is continuous for all x, when n is odd and for x > 0, when x is even. We also have ln 𝑥 is continuous for x > 0 and 𝑠𝑖𝑛−1(𝑥) and 𝑐𝑜𝑠−1(𝑥) for -1<x<1.
Example: “Fixing” a FunctionDetermine values of a and b that make the given function continuous.
𝑓 𝑥 =
2𝑠𝑖𝑛𝑥
𝑥𝑖𝑓 𝑥 < 0
𝑎 𝑖𝑓 𝑥 = 0𝑏𝑐𝑜𝑠 𝑥 𝑖𝑓 𝑥 > 0
Properties of Continuous Functions
If the functions and are continuous at , then the
following combinations are continuous at .
1. Su ms:
2. Differences:
3. Products:
4. Constant multiples: , for any number
5. Quotients: , pr
f g x c
x c
f g
f g
f g
k f k
f
g
ovided 0g c
Example: Continuity for a Rational Function
Determine where f is continuous, for 𝑓 𝑥 =𝑥4−3𝑥2+2
𝑥2−3𝑥−4.
Think about why you don’t see anything peculiar about the graph at x = -1.
Composite of Continuous Functions
If is continuous at and is continuous at , then the
composite is continuous at .
f c g f c
g f c
Intermediate Value Theorem for Continuous Functions***** STAR STAR STAR STAR STAR *****
0 0
A function that is continuous on a closed interval [ , ]
takes on every value between and . In other words,
if is between and , then for some in [ , ].
y f x a b
f a f b
y f a f b y f c c a b
Note: While this result seems reasonable, the proof is a lot more complicated than you might think, and it waits for a more advanced Calculus course.
Example: Intermediate Value Theorem
Show that 𝑓 𝑥 = 𝑥4 + 𝑥 − 3 has a root between -2 and 0 and another one between 0 and 2.