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We investigate what continuity means for functions of several variables.

This section investigates what it means for functions to be “continuous.” We begin with a series of definitions. We are used to “open intervals” such as $(1,3)$, which represents the set of all $x$ such that $1, and “closed intervals” such as $[1,3]$, which represents the set of all $x$ such that $1\leq x\leq 3$. We need analogous definitions for open and closed sets in $\R ^n$.

### Limits

On to the definition of a limit!

Suppose that $F:\R ^2\to \R$, $\vec {x} = \vector {x,y}$, and $\vec {a} = \vector {a,b}$. What do we write for $\lim _{\vec {x}\to \vec {a}} F(\vec {x}) = L$?
Suppose that $F:\R ^3\to \R$, $\vec {x} = \vector {x,y,z}$, and $\vec {a} = \vector {a,b,c}$. What do we write for $\lim _{\vec {x}\to \vec {a}} F(\vec {x}) = L$?

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 $\arctan (y/x)$,

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 $y=x^2$ gives a limit of $3$, and approaching along the parabola $y=-x^2$ gives a limit of $-3$. Thus this is a case where the limit does not exist.

### Continuity

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

To really use this definition, we need limit laws which in some sense are really continuity laws.

True or false: If $F:\R ^2\to \R$ and $G:\R ^2\to \R$ are continuous functions on an open disk $B$, then $F\pm G$ is continuous on $B$.
True False
True or false: If $F:\R ^2\to \R$ and $G:\R ^2\to \R$ are continuous functions on an open disk $B$, then $F/G$ is continuous on $B$.
True False

#### 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 $L$ if and only if $F$ approaches $L$ when $x$ approaches $a$ from either direction.

In $\R ^n$ when $n>2$ there are infinite paths from which $\vec {x}$ might approach $\vec {a}$. 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.