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

This section investigates what it means for real -valued functions of $n$-variables 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! Recall that for functions of a single variable, we say that $\lim _{x\to a} f(x) = L$ if the value of $f(x)$ can be made arbitrarily close to $L$ for all $x$ sufficiently close, but not equal to, $x=a$.

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

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 $x$ to become close to $a$; we could take $x$ to approach the point $a$ from the left or the right. For instance, tells us to consider the inputs $x only. In fact, there’s a theorem that guarantees that $\lim _{x\to a} f(x) = L$ if and only if $\lim _{x\to a^-}f(x) =L$ and $\lim _{x\to a^+}f(x) =L$, meaning that the function must approach the same value as the input approaches $a$ from both the left and the right.

On the other hand, there are now infinitely many ways for, e.g., $(x,y)\to (a,b)$; we can approach along a straight line path parallel to the $x$-axis or $y$-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.

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.

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

If it is possible to arrive at different limiting values by approaching $\vec {a}$ 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.

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.

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