Chapter 2.3

Importance of Continuity
• Before we define continuity, let’s spend a little time asking why we should care about continuity
•
From this point on, nearly every theorem you see will begin with something like “Let 𝑓 be a continuous function”
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What this means is that the truth of the theorem is conditional on the continuity of the function
•
Or stated another way, if the function is not continuous over some interval we are interested in, then we can’t use it
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Importance of Continuity
•
An imprecise but intuitionally useful way to talk about a continuous function is to say that it is a function that can be drawn without lifting pencil from paper
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The precise definition is tied to the notion of limits in a particular way
(as you’ll soon see)
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In fact, the notion of continuity arises with respect to functions because the number line itself is continuous; i.e., it has no “gaps”
•
The first example takes advantage of the intuitional view from above
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Example 1: Investigating Continuity
Find the points at which the function 𝑓 shown below is continuous and the points at which it is discontinuous.
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Example 1: Investigating Continuity
Find the points at which the function 𝑓 shown below is continuous and the points at which it is discontinuous.
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Example 1: Investigating Continuity
Find the points at which the function 𝑓 shown below is continuous and the points at which it is discontinuous.
The following are important to note about this function:
• The function “begins” at 𝑥 = 0 , lim 𝑥→0 +
•
At 𝑥 = 1 , lim 𝑥→1 + 𝑓(𝑥) = 0 but lim 𝑥→1 − 𝑓(𝑥) = 1 𝑓(𝑥) = 1
, and
, hence 𝑓 0 = 1 lim 𝑥→1 𝑓(𝑥) DNE
•
At 𝑥 = 2 , we have lim 𝑥→2 + 𝑓(𝑥) = 1 and lim 𝑥→2
− 𝑓(𝑥) = 1 so lim 𝑥→2 𝑓 2 = 2 (the limit and the function are different at 𝑥 = 2 ) 𝑓(𝑥) = 1 , however
•
At 𝑥 = 4 , lim 𝑥→4 − 𝑓(𝑥) = 1
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Conditions for Continuity
From these observations we see that:
•
A function is not continuous at a given 𝑥 if the limit does not exist there, even if the function is defined at that 𝑥
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A function is not continuous at a given 𝑥 if the limit exists, but the function value at 𝑥 is different from the limit
There are 3 conditions necessary for us to say that a function is continuous at a point
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Continuity at a Point
DEFINITION:
A function 𝑦 = 𝑓(𝑥) is continuous at an interior point 𝒄 of its domain if:
• 𝑓(𝑐) is defined
• lim 𝑥→𝑐 𝑓(𝑥) exists
• 𝑓 𝑐 = lim 𝑥→𝑐 𝑓(𝑥)
A function 𝑦 = 𝑓(𝑥) is continuous at a left endpoint 𝒂 or is continuous at a right endpoint 𝒃 of its domain if
• 𝑓(𝑎) or 𝑓(𝑏) is defined, respectively
• lim 𝑥→𝑎 + 𝑓(𝑥) exists or
• 𝑓 𝑎 = lim 𝑥→𝑎 + 𝑓(𝑥) lim 𝑥→𝑏 or
− 𝑓(𝑥) exists, respectively 𝑓 𝑏 = lim 𝑥→𝑏 − 𝑓(𝑥) , respectively
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Continuity at a Point
•
If any one of the three previous conditions fails to hold at a point 𝑐 in the domain of a function, then we say that 𝑓 is discontinuous at 𝒄
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We may also say that 𝑐 is a point of discontinuity of 𝑓
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Make sure to note that this definition refers to continuity at a single value of 𝑥
•
We will eventually want to talk about the continuity (or discontinuity) of a function either over some interval, or perhaps over its entire domain
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Example 2: Finding Points of Continuity and
Discontinuity
Find the points of continuity and the points of discontinuity of the floor function, 𝑓 𝑥 = 𝑥 .
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Example 2: Finding Points of Continuity and
Discontinuity
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Example 2: Finding Points of Continuity and
Discontinuity
Find the points of continuity and the points of discontinuity of the floor function, 𝑓 𝑥 = 𝑥 .
We can define 𝑓 as 𝑓 𝑥 = 𝑎 ∈ ℤ if 𝑎 ≤ 𝑥 < 𝑎 + 1
Now note that, for some value 𝑐 in of (𝑎, 𝑎 + 1) we have 𝑓 𝑐 = 𝑎 lim 𝑥→𝑐 𝑓(𝑥) = 𝑎 and lim 𝑥→𝑐 𝑓(𝑥) = 𝑓(𝑐)
This means that 𝑓 is continuous at every point between the integers 𝑎 and 𝑎 + 1
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Example 2: Finding Points of Continuity and
Discontinuity
Find the points of continuity and the points of discontinuity of the floor function, 𝑓 𝑥 = 𝑥 .
What about 𝑎 = 𝑐 ? We have 𝑓 𝑐 = 𝑓 𝑎 = 𝑎
But lim 𝑥→𝑐 − 𝑓(𝑥) = 𝑎 while lim 𝑥→𝑐
+ 𝑓(𝑥) = 𝑎 − 1
This means that the limit does not exist. Since this is required for continuity, then the function is discontinuous at all integer values.
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Example 2: Finding Points of Continuity and
Discontinuity
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Types of Discontinuity
•
We will encounter four types of discontinuity
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Removable discontinuities
: a function has a “hole” at one or more values of 𝑥 ; these can be removed by defining a value and “plugging” the hole
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Jump discontinuities : a function makes a sudden “jump” from function value to another at one or more values of 𝑥 (like the floor function)
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Infinite discontinuities : occurs when a function has one or more vertical asymptotes
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Oscillating discontinuities : occurs when a function begins to oscillate rapidly as 𝑥 → 𝑐 without settling near one value; we often see this with trigonometric function
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The next slide shows examples (from page 80 in your textbook)
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Types of Discontinuity
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Removable Discontinuities
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Consider the rational function 𝑥 2 + 3𝑥 + 2 𝑓 𝑥 = 𝑥 2 − 9
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The numerator and denominator can be factored to give 𝑥 − 3 𝑥 + 2 𝑥 + 1 𝑓 𝑥 = 𝑥 + 3 𝑥 − 3
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The common factors 𝑥 − 3 cancel, but this doesn’t eliminate them from the original function; 𝑓 is not defined at 𝑥 = 3 (nor at 𝑥 = −3 )
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But we can remove the discontinuity as shown in the next slide
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Removable Discontinuities
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Continuous Functions
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The previous examples, and our definition, are concerned with continuity at a single point
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We say that a function is continuous on an interval if and only if it is continuous at every point of the interval
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We say that a continuous function is one that is continuous at every point of its domain
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A continuous function need not be continuous at every interval
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Continuous Functions
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The last statement can be illustrated by considering the function 𝑦 =
1
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The domain of this function is −∞, 0 ∪ (0, ∞) and it is continuous 𝑥 at every point in this interval, so we say that it a continuous function
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However, we may also say that it is discontinuous at 𝑥 = 0
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Also, it is not continuous on any interval that contains zero, such as
[−1,1]
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Would it be correct to say that since, by definition, a function is defined at all 𝑥 -values of its domain, then it must be a continuous function? If not, can you give a counterexample?
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Continuous Functions
•
It is worthwhile to remember that all of the following are continuous functions
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Polynomial functions are continuous at all real numbers (we may also say that polynomial functions are everywhere continuous)
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Rational functions are continuous at every point of the domains and have points of discontinuity at the zeros of their denominator
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The absolute value function is everywhere continuous
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Exponential functions are everywhere continuous
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Trigonometric functions are continuous at every point of their domains, but only sine and cosine functions are everywhere continuous
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Radical functions are continuous at every point of their domains
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Properties of Continuous Functions
THEOREM:
If the function 𝑓 and 𝑔 are continuous at 𝑥 = 𝑐 , then the following combinations are continuous at 𝑥 = 𝑐
1. Sums/differences: 𝑓 ± 𝑔 is continuous at 𝑥 = 𝑐
2. Products: 𝑓𝑔 is continuous at 𝑥 = 𝑐
3. Constant multiples: 𝑘𝑓 is continuous at 𝑥 = 𝑐 , where 𝑘 is a real number
4. Quotients: 𝑓 𝑔 is continuous at 𝑥 = 𝑐 , provided that 𝑔 𝑐 ≠ 0
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Composite of Continuous Functions
THEOREM:
If 𝑓 is continuous at 𝑐 and 𝑔 is continuous at 𝑓(𝑐) , then the composite 𝑔 𝑓 𝑥 is continuous at 𝑐 .
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Example 4: Composition of Continuous
Functions
Show that 𝑓 𝑥 = 𝑥 sin 𝑥 𝑥 2 +2 is continuous.
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Example 4: Composition of Continuous
Functions
Show that 𝑓 𝑥 = 𝑥 sin 𝑥 𝑥 2 +2 is continuous.
If we let 𝑔 𝑥 = |𝑥| and ℎ 𝑥 = 𝑥 +2 theorems, since 𝑥 and sin 𝑥 are everywhere continuous, then by the Product Rule, 𝑥 sin 𝑥 𝑥 sin 𝑥
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, then 𝑓 𝑥 = 𝑔 ℎ 𝑥 . Using the previous is everywhere continuous. Since everywhere continuous. Since 𝑥 2 + 2 𝑥 2 + 2 cannot equal zero, then 𝑥 sin 𝑥 is a polynomial, then it, too, is 𝑥 2 + 2 is everywhere continuous by the Quotient Rule. Finally, since ℎ and 𝑔 are everywhere continuous, then 𝑓 𝑥 = 𝑔 ℎ 𝑥 is everywhere continuous.
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The Intermediate Value Theorem
THEOREM:
A function 𝑦 = 𝑓(𝑥) that is continuous on a closed interval [𝑎, 𝑏] takes on every value between 𝑓(𝑎) and 𝑓(𝑏) . In other words, if 𝑦
0 there must exist 𝑐 between 𝑎 and 𝑏 such that 𝑦
0 is between
= 𝑓(𝑐) .
𝑓(𝑎) and 𝑓(𝑏) , then
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The Intermediate Value Theorem
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The Intermediate Value Theorem
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Example 5: Using the IVT
Is any real number exactly 1 less than its cube?
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Example 5: Using the IVT
Is any real number exactly 1 less than its cube?
This is the same as asking if there exists a number 𝑥 such that 𝑥 = 𝑥 3 this as 𝑥 3 − 𝑥 − 1 = 0
− 1 . We can rewrite
The problem now comes down to asking if there the function 𝑓(𝑥) = 𝑥 3 root.
− 𝑥 − 1 has a real
Note that 𝑓 1 = −1 < 0 and that 𝑓 2 = 5 > 0 . That is, 𝑓 1 < 0 < 𝑓(2)
We can invoke the IVT and conclude that, since 𝑓 is continuous and since 𝑓 1 < 0 < 𝑓(2) then there must exist (at least one) number 1 < 𝑐 < 2 so that 𝑓 𝑐 = 0 . (Make sure you understand that IVT doesn’t give the number, only guarantees its existence.)
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Exercise 2.3
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