We investigate what continuity means for real-valued functions of several variables.
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 .
- 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.
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.
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.
- 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.
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.
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 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.
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 :
does not exist.
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.
Now we will use the idea of a limit to define continuity.
is continuous on an open ball if is continuous at all points in .
( by the sum law )
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 :