AP Calculus AB
Objectives:
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Class Notes
2.3 Continuity
pp 78-86
Define continuity at a point and determine where a function is and is not continuous.
Memorize and explain the test for continuity.
Differentiate between jump, infinite, oscillating and removable discontinuities.
Use algebraic combinations of functions to justify the continuity of a function.
State and apply The Intermediate Value Theorem for Continuous Functions.
Outline
What is continuity? Why is it important?
Types of discontinuities.
oscillating.
removable
infinite
jump
Intermediate Value Theorem for Continuous Functions
Vocabulary
continuity at a point
continuous at an endpoint
continuous at an interior point
discontinuous
point of discontinuity
infinite discontinuity
jump discontinuity
oscillating discontinuity
removable discontinuity
continuous extension
continuous function
continuous on an interval
Properties of Continuous Functions
connected graph
intermediate value property
Intermediate Value Theorem for ContinuousFunctions
Omissions
none
In section 2.3, will define and determine the continuity of functions.
A function is continuous at every point where you can draw the function without lifting your pencil off the paper.
This function is continuous on every point in its domain except at
x = ________ because of___________
x = ________ because of___________
Memorize this Definition!
If a function is not continuous at a point, we call that point a discontinuity. It’s important to be able to find
discontinuities of a function. Why? Well, most of what we do in calculus, we can only do to functions that are
continuous.
Types of Discontinuities
Jump discontinuity: The limit does not exist due to a “jump.”
Infinite discontinuity: The function goes to infinity or negative infinity.
Oscillating discontinuity: a function oscillates with increasing (infinite) frequency as it approaches a point
Removable discontinuity: a hole in the function that can easily be “plugged” to make the function continuous.
(Removable discontinuities occur where the function states there is one (usually meaning a piece-wise function), or
where you cancel factors from rational functions.
.
Jump Discontinuity
Infinite Discontinuity
Oscillating Discontinuity
Removable
Discontinuity*
(33-54). Find the x-value (if any) at which f is not continuous. Which of the discontinuities are removable?
33.
𝑓(𝑥) = 𝑥 2 − 2𝑥 + 1
34.
35.
37.
𝑥−3
40.
𝑓(𝑥) = 𝑥 2 −9
𝑓(𝑥) = x2 +1
41.
𝑓(𝑥) =𝑥 2 −3𝑥−10
𝑓(𝑥) = 3𝑥 − 𝑐𝑜𝑠𝑥
44.
𝑓(𝑥) =
39.
𝑓(𝑥) = 𝑥 2 +1
1
𝑥
𝑓(𝑥) = 𝑥 2 −𝑥
𝑥+2
|𝑥−3|
𝑥−3
𝑥
To see if piecewise function are continuous, examine the limits of the function as it approaches
the “interface points.”
𝑥 𝑥≤1
−2𝑥 + 3 𝑥 < 1
45.
𝑓(𝑥) = { 2
46.
𝑓(𝑥) = {
𝑥2
𝑥≥1
𝑥
𝑥>1
Find the constant(s) a (and b) such that the function is continuous on the entire real line.
3
57.
𝑓(𝑥) = { 𝑥 2
𝑎𝑥
𝑥≤2
𝑥>2
4𝑠𝑖𝑛𝑥
58.
𝑓(𝑥) = {
𝑥
𝑎 − 2𝑥
𝑥<0
𝑥≥0
59.
2,
𝑓(𝑥) = {𝑎𝑥 + 𝑏
−2
𝑥 ≤ −1
−1 < 𝑥 < 3
𝑥≥3
60.
2
2
𝑓(𝑥) = {(𝑥 − 𝑎 )/(𝑥 − 𝑎) 𝑥 ≠ 𝑎
8
𝑥=𝑎
End of Part I
The Intermediate Value
Theorem. (memorize)
The intermediate value
theorem says that
continuous functions
must take on every value
between the values of the
endpoints. This is a
somewhat self-evident
concept: If in one year you
go from 5’2” to 5’3”, your
height will be every
possible value between
5’2” and 5’3” during that
year.
The Intermediate Value Theorem is one of the 7 Fundamental Theorems you must know for AP Calculus. Here it is
in action:
Explain why the function has a zero in the given interval.
1
75.
𝑓(𝑥) = 𝑥 4 − 𝑥 3 + 3
[1, 2]
16
76.
𝑓(𝑥) = 𝑥 3 + 3𝑥 − 2
[0, 1]
𝑓(𝑥) = 𝑥 2 − 2 − 𝑐𝑜𝑠𝑥
77.
[0, π]
The Intermediate Value Theorem shows up a lot in AP Calculus. Here’s an easy problem from the 1998 AB multiple
choice.
Here’s the first part of question 3 from the 1997 AB Free Response
section.
The functions f and g are differentiable for all real numbers, and
g is strictly increasing. The table (right) gives values of the
functions and their first derivatives at selected values of x. The
function h is given by ℎ(𝑥) = 𝑓(𝑔(𝑥)) − 6.
a)
explain why there must be a value of r for 1 < r < 3 such
that ℎ(𝑟) = −5
Here’s one that down-right evil; #18 from the 1973 BC exam. It’s a doosey!