Continuity of Arc Continuity of Line

We investigate what continuity means for real-valued functions of several variables.

This section investigates what it means for real -valued functions of -variables to be "continuous" .

We begin with a series of definitions. We are used to "open intervals" such as , which represents the set of all such that , and "closed intervals" such as , which represents the set of all such that . We need analogous definitions for open and closed sets in .

We give these definitions in general, for when one is working in :

• An open ball in centered at with radius is the set of all vectors such that . In an open ball is often called an open disk.

  

• A point (denoted by a vector) in is an interior point of if there is an open ball centered at that contains only points in . We can write this in symbols as

  

• Let be a set of points in . A point (denoted by a vector) in is a boundary point of if all open balls centered at contain both points in and points not in .

  

• A set is open if every point in is an interior point.

  

• A set is closed if it contains all of its boundary points.

  

• A set is bounded if there is an open ball centered at the origin of radius such that A set that is not bounded is unbounded.

Given a set , we denote the boundary of by .

Consider a closed disk in . Describe .

Since is the boundary of a closed disk in , is a disk a circle a ball a line .

Determine if the domain of the function is open, closed, or neither, and if it is bounded.

We've already found the domain of this function to be This is the region bounded by the ellipse . Since the region includes the boundary (indicated by the use of ""), the set contains does not contain all of its boundary points and hence is closed. The region is bounded unbounded as a disk of radius , centered at the origin, contains .

Determine if the domain of is open, closed, or neither, and if it is bounded.

As we cannot divide by , we find the domain to be In other words, the domain is the set of all points not on the line . For your viewing pleasure, we have included a graph:

Note how we can draw an open disk around any point in the domain that lies entirely inside the domain, and also note how the only boundary points of the domain are the points on the line . We conclude the domain is an open set a closed set neither open nor closed set . Moreover, the set is bounded unbounded .

Limits

On to the definition of a limit! Recall that for functions of a single variable, we say that if the value of can be made arbitrarily close to for all sufficiently close, but not equal to, .

This easily allows us to make a similar definition for functions of several variables.

Suppose that is a real-valued function of variables. Assume that the domain of contains a small -dimensional ball centered at a point , except, possibly, the point . The limit of as approaches is if the value of can be made as close as one wishes to for all sufficiently close, but not equal to, .

When this occurs, we write

While the intuitive idea behind limits seems to remain unchanged, something interesting is worth observing. One of the most important ideas for limits of a function of a single variable is the notion of a sided limit. For functions of a single variable, there were really only two natural ways for to become close to ; we could take to approach the point from the left or the right. For instance, tells us to consider the inputs only. In fact, there's a theorem that guarantees that if and only if and , meaning that the function must approach the same value as the input approaches from both the left and the right.

On the other hand, there are now infinitely many ways for, e.g., ; we can approach along a straight line path parallel to the -axis or -axis, other straight line paths, or even other types of curves.

In order to check whether a limit exists, do we have to verify that the function tends to the same value along infinitely many different paths?

While this may seem problematic, there is some good news; many of the limit laws from before still do hold now.

Limit Laws Let and be real-valued functions of variables, and , and be real numbers, where Let , and . We also assume that the domain of and the domain of both contain a small ball with the center at , except possibly the point .

Constant Law

Identity Law

, where

Sum/Difference Law

Scalar Multiple Law

Product Law

Quotient Law

, if

In practice, this allows us to compute many limits in a similar fashion as before.

Compute .

We show how the above properties are used quite explicitly.

Essentially, the above laws allow us to evaluate limits by directly substituting values into the given function, provided the end result is a constant. Henceforth, when a limit can be evaluated by direct substitution, we will not show the details.

As it turns out, another old technique works well too.

Compute .

To compute this limit, note that direct substitution leads us to the indeterminate form .

What allows us to perform the cancellation of the common factors of ? Note that when determining whether a limit exists or not, we must look near the point , but not at the point . No matter how close a point is to the point , as long as , then . So, this cancellation is valid.

Limits exist when functions locally look like a smooth sheet.

When limits don't exist

When dealing with functions of a single variable we often encounter a situation where the two one-sided limits are not equal, i.e. In that case, we say that does not exist.

In when there are infinite paths along which might approach .

If it is possible to arrive at different limiting values by approaching along different paths, the limit does not exist.

This is analogous to the left and right hand limits of single variable functions not being equal, implying that the limit does not exist. Our theorems tell us that we can evaluate most limits quite simply, without worrying about paths. When indeterminate forms arise, the limit may or may not exist. The case where the limit does not exist is often easier to deal with, for we can often pick two paths along which the limit is different.

Show that does not exist by finding the limits along the lines .

Evaluating along the lines means replace all 's with and evaluating the resulting limit: While the limit exists for each choice of , we get a different limit for each choice of . Suppose , then: Now suppose that , then: Since we find differing limiting values when computing the limit along different paths, we must conclude that the limit does not exist. We finish by presenting you with a plot of :

Show that

does not exist.

Evaluating along the lines means replace all 's with and evaluating the resulting limit: Since the limit is equal to zero for each choice of , you may jump to a conclusion...but, wait! We have only examined what is happening when paths are straight lines. But, there are many other paths along which can approach . For example, consider the curve . Compute the limiting value of along this path: Since we find differing limiting values when computing the limit along different paths, we must conclude that the limit does not exist.

Here we see the function:

If one approaches the origin along any line, you see the limit is zero, by following the path on the surface. However, if one approaches the origin along a parabola, then we see the limit does not exist, as approaching along the parabola gives a limit of . Thus this is a case where the limit does not exist.

Limits don't exist when the function makes a large jump, or when the surface is somehow pinched.

Continuity

Now we will use the idea of a limit to define continuity.

Let be a real-valued function of variables, defined on the set that contains an open ball centered at . is continuous at , if

• exists.
• exists.

is continuous on an open ball if is continuous at all points in .

True or false: If and are continuous functions on an open disk , then is continuous on .

True False

Take any point, say, in . Since and are both continuous on , they are both continuous at . This means that and . This implies that

( by the sum law )

True or false: If and are continuous functions on an open disk , then is continuous on .

True False

The function may or may not be continuous, it depends on whether . If , then not continuous at that point.

Composition Limit Law Let be a continuous function on an interval . Let , where , be a function whose range is contained in (or equal to) . Then

Composition of Composite Functions Let , where , be continuous on an open disk , where the range of on is , and let be a single variable function that is continuous on . Then, the function is continuous on . In other words, for all points in .

Show that the function is continuous for all points in .

Let Since is not actually used in the function, and polynomials are continuous are not continuous , we conclude is continuous everywhere. A similar statement can be made about Setting we obtain a continuous function from . Since sine is continuous is not continuous for all real values, the composition of sine with is continuous. Hence, is continuous everywhere. We finish by presenting you with a plot of :

Let Is continuous at ? Is continuous everywhere?

To determine if is continuous at , we need to compare Applying the definition of , we see that: We now consider the limit Substituting for and in returns the indeterminate form , so we need to do more work to evaluate this limit.

Consider two related limits:

The first limit does not contain , and since is continuous, The second limit does not contain . But we know Finally, we can apply the Product Law
We have found that , so is continuous at .

A similar analysis shows that is continuous at all points in . As long as , we can evaluate the limit directly; when , a similar analysis shows that the limit is . Thus we can say that is continuous everywhere. We finish by presenting you with a plot of :

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Source: https://ximera.osu.edu/mooculus/calculus3/continuityOfFunctionsOfSeveralVariables/digInContinuity

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