For a Continuous Function F the Value F X y da Can Be Viewed as a Volume
Learning Objectives
- 2.4.1 Explain the three conditions for continuity at a point.
- 2.4.2 Describe three kinds of discontinuities.
- 2.4.3 Define continuity on an interval.
- 2.4.4 State the theorem for limits of composite functions.
- 2.4.5 Provide an example of the intermediate value theorem.
Many functions have the property that their graphs can be traced with a pencil without lifting the pencil from the page. Such functions are called continuous. Other functions have points at which a break in the graph occurs, but satisfy this property over intervals contained in their domains. They are continuous on these intervals and are said to have a discontinuity at a point where a break occurs.
We begin our investigation of continuity by exploring what it means for a function to have continuity at a point. Intuitively, a function is continuous at a particular point if there is no break in its graph at that point.
Continuity at a Point
Before we look at a formal definition of what it means for a function to be continuous at a point, let's consider various functions that fail to meet our intuitive notion of what it means to be continuous at a point. We then create a list of conditions that prevent such failures.
Our first function of interest is shown in Figure 2.32. We see that the graph of has a hole at a. In fact, is undefined. At the very least, for to be continuous at a, we need the following condition:
However, as we see in Figure 2.33, this condition alone is insufficient to guarantee continuity at the point a. Although is defined, the function has a gap at a. In this example, the gap exists because does not exist. We must add another condition for continuity at a—namely,
However, as we see in Figure 2.34, these two conditions by themselves do not guarantee continuity at a point. The function in this figure satisfies both of our first two conditions, but is still not continuous at a. We must add a third condition to our list:
Now we put our list of conditions together and form a definition of continuity at a point.
Definition
A function is continuous at a point a if and only if the following three conditions are satisfied:
- is defined
- exists
A function is discontinuous at a point a if it fails to be continuous at a.
The following procedure can be used to analyze the continuity of a function at a point using this definition.
Problem-Solving Strategy
Problem-Solving Strategy: Determining Continuity at a Point
- Check to see if is defined. If is undefined, we need go no further. The function is not continuous at a. If is defined, continue to step 2.
- Compute In some cases, we may need to do this by first computing and If does not exist (that is, it is not a real number), then the function is not continuous at a and the problem is solved. If exists, then continue to step 3.
- Compare and If then the function is not continuous at a. If then the function is continuous at a.
The next three examples demonstrate how to apply this definition to determine whether a function is continuous at a given point. These examples illustrate situations in which each of the conditions for continuity in the definition succeed or fail.
Example 2.26
Determining Continuity at a Point, Condition 1
Using the definition, determine whether the function is continuous at Justify the conclusion.
Example 2.27
Determining Continuity at a Point, Condition 2
Using the definition, determine whether the function is continuous at Justify the conclusion.
Example 2.28
Determining Continuity at a Point, Condition 3
Using the definition, determine whether the function is continuous at
Checkpoint 2.21
Using the definition, determine whether the function is continuous at If the function is not continuous at 1, indicate the condition for continuity at a point that fails to hold.
By applying the definition of continuity and previously established theorems concerning the evaluation of limits, we can state the following theorem.
Theorem 2.8
Continuity of Polynomials and Rational Functions
Polynomials and rational functions are continuous at every point in their domains.
Proof
Previously, we showed that if and are polynomials, for every polynomial and as long as Therefore, polynomials and rational functions are continuous on their domains.
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We now apply Continuity of Polynomials and Rational Functions to determine the points at which a given rational function is continuous.
Example 2.29
Continuity of a Rational Function
For what values of x is continuous?
Checkpoint 2.22
For what values of x is continuous?
Types of Discontinuities
As we have seen in Example 2.26 and Example 2.27, discontinuities take on several different appearances. We classify the types of discontinuities we have seen thus far as removable discontinuities, infinite discontinuities, or jump discontinuities. Intuitively, a removable discontinuity is a discontinuity for which there is a hole in the graph, a jump discontinuity is a noninfinite discontinuity for which the sections of the function do not meet up, and an infinite discontinuity is a discontinuity located at a vertical asymptote. Figure 2.37 illustrates the differences in these types of discontinuities. Although these terms provide a handy way of describing three common types of discontinuities, keep in mind that not all discontinuities fit neatly into these categories.
These three discontinuities are formally defined as follows:
Definition
If is discontinuous at a, then
- has a removable discontinuity at a if exists. (Note: When we state that exists, we mean that where L is a real number.)
- has a jump discontinuity at a if and both exist, but (Note: When we state that and both exist, we mean that both are real-valued and that neither take on the values ±∞.)
- has an infinite discontinuity at a if and/or
Example 2.30
Classifying a Discontinuity
In Example 2.26, we showed that is discontinuous at Classify this discontinuity as removable, jump, or infinite.
Example 2.31
Classifying a Discontinuity
In Example 2.27, we showed that is discontinuous at Classify this discontinuity as removable, jump, or infinite.
Example 2.32
Classifying a Discontinuity
Determine whether is continuous at −1. If the function is discontinuous at −1, classify the discontinuity as removable, jump, or infinite.
Checkpoint 2.23
For decide whether f is continuous at 1. If f is not continuous at 1, classify the discontinuity as removable, jump, or infinite.
Continuity over an Interval
Now that we have explored the concept of continuity at a point, we extend that idea to continuity over an interval. As we develop this idea for different types of intervals, it may be useful to keep in mind the intuitive idea that a function is continuous over an interval if we can use a pencil to trace the function between any two points in the interval without lifting the pencil from the paper. In preparation for defining continuity on an interval, we begin by looking at the definition of what it means for a function to be continuous from the right at a point and continuous from the left at a point.
Continuity from the Right and from the Left
A function is said to be continuous from the right at a if
A function is said to be continuous from the left at a if
A function is continuous over an open interval if it is continuous at every point in the interval. A function is continuous over a closed interval of the form if it is continuous at every point in and is continuous from the right at a and is continuous from the left at b. Analogously, a function is continuous over an interval of the form if it is continuous over and is continuous from the left at b. Continuity over other types of intervals are defined in a similar fashion.
Requiring that and ensures that we can trace the graph of the function from the point to the point without lifting the pencil. If, for example, we would need to lift our pencil to jump from to the graph of the rest of the function over
Example 2.33
Continuity on an Interval
State the interval(s) over which the function is continuous.
Example 2.34
Continuity over an Interval
State the interval(s) over which the function is continuous.
Checkpoint 2.24
State the interval(s) over which the function is continuous.
The Composite Function Theorem allows us to expand our ability to compute limits. In particular, this theorem ultimately allows us to demonstrate that trigonometric functions are continuous over their domains.
Theorem 2.9
Composite Function Theorem
If is continuous at L and then
Before we move on to Example 2.35, recall that earlier, in the section on limit laws, we showed Consequently, we know that is continuous at 0. In Example 2.35 we see how to combine this result with the composite function theorem.
Example 2.35
Limit of a Composite Cosine Function
Evaluate
Checkpoint 2.25
Evaluate
The proof of the next theorem uses the composite function theorem as well as the continuity of and at the point 0 to show that trigonometric functions are continuous over their entire domains.
Theorem 2.10
Continuity of Trigonometric Functions
Trigonometric functions are continuous over their entire domains.
Proof
We begin by demonstrating that is continuous at every real number. To do this, we must show that for all values of a.
The proof that is continuous at every real number is analogous. Because the remaining trigonometric functions may be expressed in terms of and their continuity follows from the quotient limit law.
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As you can see, the composite function theorem is invaluable in demonstrating the continuity of trigonometric functions. As we continue our study of calculus, we revisit this theorem many times.
The Intermediate Value Theorem
Functions that are continuous over intervals of the form where a and b are real numbers, exhibit many useful properties. Throughout our study of calculus, we will encounter many powerful theorems concerning such functions. The first of these theorems is the Intermediate Value Theorem.
Theorem 2.11
The Intermediate Value Theorem
Let f be continuous over a closed, bounded interval If z is any real number between and then there is a number c in satisfying in Figure 2.38.
Example 2.36
Application of the Intermediate Value Theorem
Show that has at least one zero.
Example 2.37
When Can You Apply the Intermediate Value Theorem?
If is continuous over and can we use the Intermediate Value Theorem to conclude that has no zeros in the interval Explain.
Example 2.38
When Can You Apply the Intermediate Value Theorem?
For and Can we conclude that has a zero in the interval
Checkpoint 2.26
Show that has a zero over the interval
Section 2.4 Exercises
For the following exercises, determine the point(s), if any, at which each function is discontinuous. Classify any discontinuity as jump, removable, infinite, or other.
131.
132 .
133.
134 .
135.
136 .
137.
138 .
For the following exercises, decide if the function continuous at the given point. If it is discontinuous, what type of discontinuity is it?
139.
at
140 .
at
141.
at
142 .
at
143.
at
144 .
at
In the following exercises, find the value(s) of k that makes each function continuous over the given interval.
145.
146 .
147.
148 .
149.
In the following exercises, use the Intermediate Value Theorem (IVT).
150 .
Let Over the interval there is no value of x such that although and Explain why this does not contradict the IVT.
151.
A particle moving along a line has at each time t a position function which is continuous. Assume and Another particle moves such that its position is given by Explain why there must be a value c for such that
152 .
[T] Use the statement "The cosine of t is equal to t cubed."
- Write a mathematical equation of the statement.
- Prove that the equation in part a. has at least one real solution.
- Use a calculator to find an interval of length 0.01 that contains a solution.
153.
Apply the IVT to determine whether has a solution in one of the intervals or Briefly explain your response for each interval.
154 .
Consider the graph of the function shown in the following graph.
- Find all values for which the function is discontinuous.
- For each value in part a., state why the formal definition of continuity does not apply.
- Classify each discontinuity as either jump, removable, or infinite.
155.
Let
- Sketch the graph of f.
- Is it possible to find a value k such that which makes continuous for all real numbers? Briefly explain.
156 .
Let for
- Sketch the graph of f.
- Is it possible to find values and such that and and that makes continuous for all real numbers? Briefly explain.
157.
Sketch the graph of the function with properties i. through vi.
- The domain of f is
- f has an infinite discontinuity at
- f is left continuous but not right continuous at
- and
158 .
Sketch the graph of the function with properties i. through iv.
- The domain of f is
- and exist and are equal.
- is left continuous but not continuous at and right continuous but not continuous at
- has a removable discontinuity at a jump discontinuity at and the following limits hold: and
In the following exercises, suppose is defined for all x. For each description, sketch a graph with the indicated property.
159.
Discontinuous at with and
160 .
Discontinuous at but continuous elsewhere with
Determine whether each of the given statements is true. Justify your response with an explanation or counterexample.
161.
is continuous everywhere.
162 .
If the left- and right-hand limits of as exist and are equal, then f cannot be discontinuous at
163.
If a function is not continuous at a point, then it is not defined at that point.
164 .
According to the IVT, has a solution over the interval
165.
If is continuous such that and have opposite signs, then has exactly one solution in
166 .
The function is continuous over the interval
167.
If is continuous everywhere and then there is no root of in the interval
[T] The following problems consider the scalar form of Coulomb's law, which describes the electrostatic force between two point charges, such as electrons. It is given by the equation where is Coulomb's constant, are the magnitudes of the charges of the two particles, and r is the distance between the two particles.
168 .
To simplify the calculation of a model with many interacting particles, after some threshold value we approximate F as zero.
- Explain the physical reasoning behind this assumption.
- What is the force equation?
- Evaluate the force F using both Coulomb's law and our approximation, assuming two protons with a charge magnitude of and the Coulomb constant are 1 m apart. Also, assume How much inaccuracy does our approximation generate? Is our approximation reasonable?
- Is there any finite value of R for which this system remains continuous at R?
169.
Instead of making the force 0 at R, instead we let the force be 10−20 for Assume two protons, which have a magnitude of charge and the Coulomb constant Is there a value R that can make this system continuous? If so, find it.
Recall the discussion on spacecraft from the chapter opener. The following problems consider a rocket launch from Earth's surface. The force of gravity on the rocket is given by where m is the mass of the rocket, d is the distance of the rocket from the center of Earth, and k is a constant.
170 .
[T] Determine the value and units of k given that the mass of the rocket is 3 million kg. (Hint: The distance from the center of Earth to its surface is 6378 km.)
171.
[T] After a certain distance D has passed, the gravitational effect of Earth becomes quite negligible, so we can approximate the force function by Using the value of k found in the previous exercise, find the necessary condition D such that the force function remains continuous.
172 .
As the rocket travels away from Earth's surface, there is a distance D where the rocket sheds some of its mass, since it no longer needs the excess fuel storage. We can write this function as Is there a D value such that this function is continuous, assuming
Prove the following functions are continuous everywhere
173.
174 .
175.
Where is continuous?
Source: https://openstax.org/books/calculus-volume-1/pages/2-4-continuity
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