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**.

**boundary**of by .

*bounded*by the ellipse . Since the region includes the boundary (indicated by the use of “”), the set containsdoes not contain all of its boundary points and hence is closed. The region is boundedunbounded as a disk of radius , centered at the origin, contains .

*not*on the line . For your viewing pleasure, we have included a graph:

### Limits

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 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.

### Continuity

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

**continuous**at , if

- exists.
- exists.

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*.

where

- 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 limits don’t exist

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 :