Download Continuity & One

Continuity & One-Sided
Limits
Chapter 2.4
Continuity at a Point & on an Open Interval
• The mathematical meaning of continuity is not too different from the
everyday meaning: something is continuous if it has no breaks or
interruptions
• Beginning with Chapter 3, nearly every theorem and definition will
include as one condition that a function be continuous
• Thus continuity is an extremely important concept
Definition of Continuity
DEFINITION:
Continuity at a point
A function f is continuous at c if the following three conditions are met
1. 𝑓(𝑐) is defined
2.
3.
lim 𝑓(𝑥) exists
𝑥→𝑐
lim 𝑓(𝑥) = 𝑓(𝑐) (i.e., the limit of the function equals the value of the function at c)
𝑥→𝑐
Continuity on an open interval
A function is continuous on an open interval (𝒂, 𝒃) if it is continuous at each point in the interval. A function
that is continuous on the entire real line is everywhere continuous.
Continuity at a Point & on an Open Interval
Continuity at a Point & an Open Interval
• Discontinuities can be either removable or non-removable
• A discontinuity is removable if it can be made continuous by defining
or redefining 𝑓(𝑐)
Example 1: Continuity of a Function
Discuss the continuity of each function
a)
𝑓 𝑥 =
b) 𝑔 𝑥 =
c)
1
𝑥
𝑥 2 −1
𝑥−1
𝑥 + 1, 𝑥 ≤ 0
ℎ 𝑥 =
𝑒 𝑥, 𝑥 > 0
d) 𝑦 = sin 𝑥
Example 1: Continuity of a Function
a) The function is defined everywhere except 𝑥 = 0. It is continuous for every x in its domain. It has a nonremovable discontinuity at 𝑥 = 0.
b) The function is a line with a hole at 𝑥 = 1. Thus it has a removable discontinuity at 𝑥 = 1. We could
remove it by redefining the function as
𝑥2 − 1
𝑔 𝑥 = 𝑥 − 1 ,𝑥 ≠ 1
2, 𝑥 = 1
c) Note that at 𝑥 = 0, the function is defined: ℎ 0 = 0 + 1 = 1. The exponential part of the function does not
include zero, but lim 𝑒 𝑥 = 1, so all three conditions are met and the function is everywhere continuous
𝑥→0
d) The function is everywhere continuous.
One-Sided Limits & Continuity on a Closed
Interval
DEFINITION:
The notation
lim 𝑓(𝑥) = 𝐿
𝑥→𝑐 +
means that x approaches c from the right, that is, from values 𝑥 > 𝑐. We say that this is the limit from the
right.
Similarly, the notation
lim 𝑓(𝑥) = 𝐿
𝑥→𝑐 −
means that x approaches c from the left, that is, from values 𝑥 < 𝑐. We say that this is the limit from the left.
Example 2: A One-Sided Limit
Find the limit of 𝑓 𝑥 = 4 − 𝑥 2 as x approaches −2 from the right.
In this case, there is no difference, other than notation, in taking a one-sided limit. We can use direct
substitution here to get
lim + 4 − 𝑥 2 = 4 − −2 2 = 0
𝑥→−2
This will differ somewhat when taking one-sided limits for piecewise functions and for step-functions, as in the
next example.
Example 3: The Greatest Integer Function
Find the limit of the greatest integer function 𝑓 𝑥 = 𝑥 as x
approaches 0 from the left and from the right.
The greatest integer function returns values of 𝑓(𝑥) that are the greatest integer less than or equal to x. For
example, 𝑓 0.325 = 0 because 0 < 0.325; 𝑓 2.103 = 2 because 2 < 2.103. This is sometimes called the
floor function.
Example 3: The Greatest Integer Function
As the values of 𝑥 > 0 approach from the right, the functions values approach zero. But as the values 𝑥 < 0
approach from the left, the function values approach −1. Hence this function has a jump discontinuity at 𝑥 = 0.
In fact, the function has discontinuities at every integer n.
Theorem 2.10: Existence of a Limit
THEOREM:
Let f be a function and let c and L be real numbers. The limit of 𝑓(𝑥) as x approaches c is L if and only if
lim 𝑓(𝑥) = 𝐿 and lim+ 𝑓(𝑥) = 𝐿
𝑥→𝑐 −
𝑥→𝑐
Definition of Continuity on a Closed Interval
• In the original definition of continuity, we defined continuity on an
open interval as being continuous at every point in the interval
• That is, we don’t have to worry about the endpoints since they are not
in the interval
• A closed interval includes its endpoints, so we must define what it
means for a function to be continuous on a closed interval
• We can use one-sided limits to do this
Definition of Continuity on a Closed Interval
DEFINITION:
A function f is continuous on the closed interval [𝒂, 𝒃] if it is continuous on the open interval (𝑎, 𝑏) and
lim 𝑓(𝑥) = 𝑓 𝑎 and lim− 𝑓(𝑥) = 𝑓(𝑏)
𝑥→𝑎+
𝑥→𝑏
The function f is continuous from the right at a and continuous from the left at b.
Similar definitions can be made to cover half-open intervals.
Example 4: Continuity on a Closed Interval
Discuss the continuity of 𝑓 𝑥 = 1 − 𝑥 2 .
The graph of f is a semicircle centered at zero with radius 1. So the interval to consider is [−1,1]. Note that f is
continuous in the open interval (−1,1), as required by the definition. Additionally we have
lim + 1 − 𝑥 2 =
𝑥→−1
1 − −1
2
= 0 and lim− 1 − 𝑥 2 =
𝑥→1
1 − 12 = 0
So by definition, we are justified in saying that f is continuous on the closed interval [−1,1].
Theorem 2.11: Properties of Continuity
THEOREM:
If b is a real number and f and g are continuous at 𝑥 = 𝑐, then the following functions are also continuous at c.
1. Scalar multiple: 𝑏𝑓
2. Sum and difference: 𝑓 ± 𝑔
3. Product: 𝑓𝑔
𝑓
4. Quotient: 𝑔 , 𝑔 𝑐 ≠ 0
Note that these follow directly from the limit properties of theorem 2.2.
Properties of Continuity
The following types of functions are continuous at every point in their
domain.
1. Polynomial: 𝑝 𝑥 = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎1 𝑥 + 𝑎0
𝑝 𝑥
𝑥
2. Rational: 𝑟 𝑥 = 𝑞
3. Radical: 𝑓 𝑥 =
𝑛
,𝑞 𝑥 ≠ 0
𝑥
4. Trigonometric: sin 𝑥 , cos 𝑥 , tan 𝑥 , csc 𝑥 , sec 𝑥 , cot 𝑥
5. Exponential & logarithmic: 𝑓 𝑥 = 𝑎 𝑥 , 𝑓 𝑥 = 𝑒 𝑥 , 𝑓 𝑥 = ln 𝑥
Example 6: Applying Properties of Continuity
By Theorem 2.11, if follows that each of the following functions is
continuous at every point in its domain.
𝑓 𝑥 = 𝑥 + 𝑒 𝑥 : the polynomial function 𝑦 = 𝑥 and the exponential function 𝑦 = 𝑒 𝑥 are everywhere continuous,
so by theorem 2.11 #2, their sum is everywhere continuous
𝑓 𝑥 = 3 tan 𝑥: by theorem 2.11 #1, the function is continuous throughout its domain (not everywhere
continuous)
𝑥 2 +1
:
cos 𝑥
𝑓 𝑥 =
the numerator is a polynomial function and the denominator is a trigonometric function, each
continuous everywhere, so by theorem 2.11 #4, f is continuous at all points for which cos 𝑥 ≠ 0
Theorem 2.12: Continuity of a Composite
Function
THEOREM:
If g is continuous at c and f is continuous at 𝑔(𝑐), then the composite function given by
𝑓 ∘ 𝑔 𝑥 = 𝑓(𝑔 𝑥 )
is continuous at c.
Note that this theorem is a direct consequence of theorem 2.5, limits of composite functions.
Example 7: Testing for Continuity
Describe the interval(s) on which each function is continuous.
a) 𝑓 𝑥 = tan 𝑥
b) 𝑔 𝑥 =
c) ℎ 𝑥 =
sin
1
𝑥
,𝑥 ≠ 0
0, 𝑥 = 0
𝑥 sin
1
𝑥
,𝑥 ≠ 0
0, 𝑥 = 0
Example 7: Testing for Continuity
𝜋
a) Since the tangent function is not defined at 𝑥 = 2 + 𝜋𝑛, n an integer, it is continuous over all intervals
𝑝𝑖
𝜋
− + 𝜋𝑛, + 𝜋𝑛 , 𝑛 an integer
2
2
b) This is the oscillating function that has no limit as x approaches zero. It is continuous at all values 𝑥 ≠ 0,
but defining the function for 𝑥 = 0 does not make it everywhere continuous the limit does not exist. Hence,
it is continuous over −∞, 0 ∪ (0, ∞).
c) This is the function from part b), but multiplied by x, which produces a “damping” effect. The function 𝑦 =
1
𝑥 sin 𝑥 is still not defined at zero, but the limit exists as x approaches zero. To see why, see the next slide
Example 7: Testing for Continuity
The function f is less than |𝑥| and greater than −|𝑥|
for values near 𝑥 = 0. That is
1
− 𝑥 ≤ 𝑥 sin
≤ |𝑥|
𝑥
Since both positive and negative absolute value
functions approach zero as x approaches zero, and
since f lies between them, we can invoke the Squeeze
Theorem to conclude that
1
lim 𝑥 sin
=0
𝑥→0
𝑥
Finally, the piecewise function defines f at 𝑥 = 0, so
the function is everywhere continuous.
The Intermediate Value Theorem
• Behind the idea of continuity lies an important principle known as the
Completeness Axiom
• Informally, the Completeness Axiom says that there are no gaps or
holes in the number line
• It would be fair to say that the Completeness Axiom creates the
continuity of the number line
• To see why, let’s take a brief look at how mathematicians developed
the concept of number and of the number line during the 19th century
The Intermediate Value Theorem
• Beginning with a few assumptions (axioms), including the existence of
a single number, 1, mathematicians were able to “create” (or logically
derive), along with their familiar properties
• The natural numbers (or counting numbers, 1, 2, 3, …)
• The integers, logically derived from the natural numbers
• The rational numbers, logically derived from the integers
• If they had stopped there, the number line would have an infinite
number of gaps in it
• Imagine that you cut the number line at a point to produce two pieces
• The cut is made at some point c so that 𝑐 2 < 2 for all numbers in the
left piece, and 𝑐 2 > 2 for all numbers in the right piece
The Intermediate Value Theorem
• What can we say about the point at c?
• In fact, we can see that if 𝑐 2 < 2, then 𝑐 < 2, so all the numbers in
the left piece of the line are numbers less than 2
• But it can be shown that 2 is not a rational number!
• So if the rational numbers were all that existed, there would be gaps,
not only at 2, but at the square roots of all numbers that are not
perfect squares, as well as at any number than cannot be written as the
ratio of two integers (i.e., the irrational numbers)
The Intermediate Value Theorem
• The key point here is that it is not possible to logically derive the
irrational numbers without the Completeness Axiom
• Without this assumption, limits would not exist for an infinite number
of points on the number line
• For example,
lim 𝑥
𝑥→2
would have no answer because of the gap
• Calculus would not be possible
The Intermediate Value Theorem
• The Intermediate Value Theorem (IVT) is a consequence of the
continuity of the number line
• From this theorem we will later derive the Mean Value Theorem,
which leads to the most important theorem in calculus, The
Fundamental Theorem of Calculus
The Intermediate Value Theorem
THEOREM:
If f is continuous on the closed interval [𝑎, 𝑏] and k is any number between 𝑓(𝑎) and 𝑓(𝑏), then there is at least
one number c in [𝑎, 𝑏] such that
𝑓 𝑐 =𝑘
Note that the conditions for this theorem to apply are that f be continuous on a closed interval, and k is a number
in the range of the function.
The Intermediate Value Theorem
• Your textbook uses a good analogy to help you understand IVT
• If a girl is 5 feet tall on her 13th birthday and 5 feet 7 inches tall on her
14th birthday, then at some point between birthdays she must have
been, say, 5 feet 3 inches tall (assuming growth is continuous)
• Humans don’t “jump” from 5 feet to 5 feet 7 inches in an instant of
time (this would be like a jump discontinuity)
The Intermediate Value Theorem
Example 8: An Application of the
Intermediate Value Theorem
Use the IVT to show that the polynomial function 𝑓 𝑥 = 𝑥 3 + 2𝑥 − 1
has at least one zero in the interval [0,1].
Look first at the values 𝑓 0 = −1 and 𝑓 1 = 2. Since all polynomial functions are everywhere continuous,
and since 𝑓 𝑎 ≤ 0 ≤ 𝑓 𝑏 ⟺ −1 ≤ 0 ≤ 2, then by the IVT we conclude that there exists at least one number
𝑐 ∈ [𝑎, 𝑏] such that
𝑓 𝑐 =0
Therefore, f has at least one zero between 0 and 1.
Note that the IVT doesn’t tell us how to find the value c; it only guarantees its existence.
Exercise 2.4a
• Page 98, #1-6, 7-27 odds, 29-36
Exercise 2.4b
• Page 99, #39-60 multiples of 3, 61-70, 71-78, 83-86, 91-94, 115-116