We investigate what continuity means for functions of several variables.
- 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!
the limit of as approaches is ,
written if the value of can be made as close as one wishes to for all sufficiently close, but not equal to, .
Limits exist when functions locally look like a smooth sheet. On the other hand, limits don’t exist when the function make a large jump, as in the case with ,
or when the surface is somehow pinched. Here we see the function:
If one approaches the origin along any line, you see the limit of the (composite) function is zero, by following the path on the surface. However, if one approaches the the origin along a parabola, then we see the limit does not exist, as approaching along the parabola gives a limit of , and approaching along the parabola gives a limit of . Thus this is a case where the limit does not exist.
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 .
To really use this definition, we need limit laws which in some sense are really continuity laws.
- Constant Law
- Identity Law
- Sum/Difference Law
- Scalar Multiple Law
- Product Law
- Quotient Law
- , if .
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 know that we can combine these two limits so that
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 :
When dealing with functions of a single variable we often considered one-sided limits and stated if and only if That is, the limit is if and only if approaches when approaches from either direction.
In when there are infinite paths from which might approach . Now we have a fact, if and only if along every path.
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. If it does exist, it can be difficult to prove this as we need to show the same limiting value is obtained regardless of the path chosen. 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 :
By applying L’Hôpital’s Rule, we can show this limit is except when , that is, along the line . This line is not in the domain of , so we have found the following fact: along every line in the domain of , Now consider the limit along the path :
Now apply L’Hôpital’s Rule twice to find a limit is of the form . Hence the limit does not exist. Step back and consider what we have just discovered.
- Along any line in the domain of the , the limit is .
- However, along the path , which lies in the domain of the for all , the limit does not exist.
Since the limit is not the same along every path to , we say does not exist. We finish by presenting you with a plot of :