Continuity of a Two Variable Function
Learning Objectives
- 4.2.1 Calculate the limit of a function of two variables.
- 4.2.2 Learn how a function of two variables can approach different values at a boundary point, depending on the path of approach.
- 4.2.3 State the conditions for continuity of a function of two variables.
- 4.2.4 Verify the continuity of a function of two variables at a point.
- 4.2.5 Calculate the limit of a function of three or more variables and verify the continuity of the function at a point.
We have now examined functions of more than one variable and seen how to graph them. In this section, we see how to take the limit of a function of more than one variable, and what it means for a function of more than one variable to be continuous at a point in its domain. It turns out these concepts have aspects that just don't occur with functions of one variable.
Limit of a Function of Two Variables
Recall from The Limit of a Function the definition of a limit of a function of one variable:
Let be defined for all in an open interval containing Let be a real number. Then
if for every there exists a such that if for all in the domain of then
Before we can adapt this definition to define a limit of a function of two variables, we first need to see how to extend the idea of an open interval in one variable to an open interval in two variables.
Definition
Consider a point A disk centered at point is defined to be an open disk of radius centered at point —that is,
as shown in the following graph.
The idea of a disk appears in the definition of the limit of a function of two variables. If is small, then all the points in the disk are close to This is completely analogous to being close to in the definition of a limit of a function of one variable. In one dimension, we express this restriction as
In more than one dimension, we use a disk.
Definition
Let be a function of two variables, and The limit of as approaches is written
if for each there exists a small enough such that for all points in a disk around except possibly for itself, the value of is no more than away from (Figure 4.15). Using symbols, we write the following: For any there exists a number such that
Proving that a limit exists using the definition of a limit of a function of two variables can be challenging. Instead, we use the following theorem, which gives us shortcuts to finding limits. The formulas in this theorem are an extension of the formulas in the limit laws theorem in The Limit Laws.
Theorem 4.1
Limit laws for functions of two variables
Let and be defined for all in a neighborhood around and assume the neighborhood is contained completely inside the domain of Assume that and are real numbers such that and and let be a constant. Then each of the following statements holds:
Constant Law:
(4.2)
Identity Laws:
(4.3)
(4.4)
Sum Law:
(4.5)
Difference Law:
(4.6)
Constant Multiple Law:
(4.7)
Product Law:
(4.8)
Quotient Law:
(4.9)
Power Law:
(4.10)
for any positive integer
Root Law:
(4.11)
for all if is odd and positive, and for if is even and positive provided that for all in neighborhood of .
The proofs of these properties are similar to those for the limits of functions of one variable. We can apply these laws to finding limits of various functions.
Example 4.8
Finding the Limit of a Function of Two Variables
Find each of the following limits:
Checkpoint 4.6
Evaluate the following limit:
Since we are taking the limit of a function of two variables, the point is in and it is possible to approach this point from an infinite number of directions. Sometimes when calculating a limit, the answer varies depending on the path taken toward If this is the case, then the limit fails to exist. In other words, the limit must be unique, regardless of path taken.
Example 4.9
Limits That Fail to Exist
Show that neither of the following limits exist:
Checkpoint 4.7
Show that
does not exist.
Interior Points and Boundary Points
To study continuity and differentiability of a function of two or more variables, we first need to learn some new terminology.
Definition
Let S be a subset of (Figure 4.17).
- A point is called an interior point of if there is a disk centered around contained completely in
- A point is called a boundary point of if every disk centered around contains points both inside and outside
Definition
Let S be a subset of (Figure 4.17).
- is called an open set if every point of is an interior point.
- is called a closed set if it contains all its boundary points.
An example of an open set is a disk. If we include the boundary of the disk, then it becomes a closed set. A set that contains some, but not all, of its boundary points is neither open nor closed. For example if we include half the boundary of a disk but not the other half, then the set is neither open nor closed.
Definition
Let S be a subset of (Figure 4.17).
- An open set is a connected set if it cannot be represented as the union of two or more disjoint, nonempty open subsets.
- A set is a region if it is open, connected, and nonempty.
The definition of a limit of a function of two variables requires the disk to be contained inside the domain of the function. However, if we wish to find the limit of a function at a boundary point of the domain, the is not contained inside the domain. By definition, some of the points of the are inside the domain and some are outside. Therefore, we need only consider points that are inside both the disk and the domain of the function. This leads to the definition of the limit of a function at a boundary point.
Definition
Let be a function of two variables, and and suppose is on the boundary of the domain of Then, the limit of as approaches is written
if for any there exists a number such that for any point inside the domain of and within a suitably small distance positive of the value of is no more than away from (Figure 4.15). Using symbols, we can write: For any there exists a number such that
Example 4.10
Limit of a Function at a Boundary Point
Prove
Checkpoint 4.8
Evaluate the following limit:
Continuity of Functions of Two Variables
In Continuity, we defined the continuity of a function of one variable and saw how it relied on the limit of a function of one variable. In particular, three conditions are necessary for to be continuous at point
- exists.
- exists.
These three conditions are necessary for continuity of a function of two variables as well.
Definition
A function is continuous at a point in its domain if the following conditions are satisfied:
- exists.
- exists.
Example 4.11
Demonstrating Continuity for a Function of Two Variables
Show that the function is continuous at point
Checkpoint 4.9
Show that the function is continuous at point
Continuity of a function of any number of variables can also be defined in terms of delta and epsilon. A function of two variables is continuous at a point in its domain if for every there exists a such that, whenever it is true, This definition can be combined with the formal definition (that is, the epsilon–delta definition) of continuity of a function of one variable to prove the following theorems:
Theorem 4.2
The Sum of Continuous Functions Is Continuous
If is continuous at and is continuous at then is continuous at
Theorem 4.3
The Product of Continuous Functions Is Continuous
If is continuous at and is continuous at then is continuous at
Theorem 4.4
The Composition of Continuous Functions Is Continuous
Let be a function of two variables from a domain to a range Suppose is continuous at some point and define Let be a function that maps to such that is in the domain of Last, assume is continuous at Then is continuous at as shown in the following figure.
Let's now use the previous theorems to show continuity of functions in the following examples.
Example 4.12
More Examples of Continuity of a Function of Two Variables
Show that the functions and are continuous everywhere.
Checkpoint 4.10
Show that the functions and are continuous everywhere.
Functions of Three or More Variables
The limit of a function of three or more variables occurs readily in applications. For example, suppose we have a function that gives the temperature at a physical location in three dimensions. Or perhaps a function can indicate air pressure at a location at time How can we take a limit at a point in What does it mean to be continuous at a point in four dimensions?
The answers to these questions rely on extending the concept of a disk into more than two dimensions. Then, the ideas of the limit of a function of three or more variables and the continuity of a function of three or more variables are very similar to the definitions given earlier for a function of two variables.
Definition
Let be a point in Then, a ball in three dimensions consists of all points in lying at a distance of less than from —that is,
To define a ball in higher dimensions, add additional terms under the radical to correspond to each additional dimension. For example, given a point in a ball around can be described by
To show that a limit of a function of three variables exists at a point it suffices to show that for any point in a ball centered at the value of the function at that point is arbitrarily close to a fixed value (the limit value). All the limit laws for functions of two variables hold for functions of more than two variables as well.
Example 4.13
Finding the Limit of a Function of Three Variables
Find
Checkpoint 4.11
Find
Section 4.2 Exercises
For the following exercises, find the limit of the function.
60 .
61.
62 .
Show that the limit exists and is the same along the paths: and and along
For the following exercises, evaluate the limits at the indicated values of If the limit does not exist, state this and explain why the limit does not exist.
63.
64 .
65.
66 .
67.
68 .
69.
70 .
71.
72 .
73.
74 .
75.
76 .
77.
For the following exercises, complete the statement.
78 .
A point in a plane region is an interior point of if _________________.
79.
A point in a plane region is called a boundary point of if ___________.
For the following exercises, use algebraic techniques to evaluate the limit.
80 .
81.
82 .
83.
For the following exercises, evaluate the limits of the functions of three variables.
84 .
85.
For the following exercises, evaluate the limit of the function by determining the value the function approaches along the indicated paths. If the limit does not exist, explain why not.
86 .
- Along the
- Along the
- Along the path
87.
Evaluate using the results of previous problem.
88 .
- Along the x-axis
- Along the y-axis
- Along the path
89.
Evaluate using the results of previous problem.
Discuss the continuity of the following functions. Find the largest region in the in which the following functions are continuous.
90 .
91.
92 .
93.
For the following exercises, determine the region in which the function is continuous. Explain your answer.
94 .
95.
(Hint: Show that the function approaches different values along two different paths.)
96 .
97.
Determine whether is continuous at
98 .
Create a plot using graphing software to determine where the limit does not exist. Find where in the coordinate plane is continuous.
99.
Determine the region of the in which the function is continuous. Use technology to support your conclusion.
100 .
Determine the region of the in which is continuous. Use technology to support your conclusion. (Hint: Choose the range of values for carefully!)
101.
At what points in space is continuous?
102 .
At what points in space is continuous?
103.
Show that does not exist at by plotting the graph of the function.
104 .
[T] Evaluate by plotting the function using a CAS. Determine analytically the limit along the path
105.
[T]
- Use a CAS to draw a contour map of
- What is the name of the geometric shape of the level curves?
- Give the general equation of the level curves.
- What is the maximum value of
- What is the domain of the function?
- What is the range of the function?
106 .
True or False: If we evaluate along several paths and each time the limit is we can conclude that
107.
Use polar coordinates to find You can also find the limit using L'Hôpital's rule.
108 .
Use polar coordinates to find
109.
Discuss the continuity of where and
110 .
Given find
111.
Given find
Source: https://openstax.org/books/calculus-volume-3/pages/4-2-limits-and-continuity
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