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To develop calculus for functions of one variable, we needed to make sense of the concept of a limit, which was used in the definition of a continuous function and the derivative of a function. Limits involving functions of two variables can be considerably more difficult to deal with; fortunately, most of the functions we encounter are fairly easy to understand.

The potential difficulty is largely due to the fact that there are many ways to 鈥渁pproach鈥� a point in the \(x\)-\(y\)-plane. If we want to say that

we need to capture the idea that as \((x,y)\) gets close to \((a,b)\) then \(f(x,y)\) gets close to \(L\text{.}\) For functions of one variable, \(f(x)\text{,}\) there are only two ways that \(x\) can approach \(a\text{:}\) from the left or right. But there are an infinite number of ways to approach \((a,b)\text{:}\) along any one of an infinite number of straight lines, or even along a curved path in the \(x\)-\(y\)-plane. We might hope that it's really not so bad鈥攕uppose, for example, that along every possible line through \((a,b)\) the value of \(f(x,y)\) gets close to \(L\text{;}\) surely this means that 鈥淺(f(x,y)\) approaches \(L\) as \((x,y)\) approaches \((a,b)\)鈥�. Sadly, no.

Fortunately, we can define the concept of limit without needing to specify how a particular point is approached鈥攊ndeed, in 顿别蹿颈苍颈迟颈辞苍听3.4, we didn't need the concept of 鈥渁pproach.鈥� Roughly, that definition says that when \(x\) is close to \(a\) then \(f(x)\) is close to \(L\text{;}\) there is no mention of 鈥渉ow鈥� we get close to \(a\text{.}\) We can adapt that definition to two variables quite easily:

This says that we can make \(|f(x,y)-L|\lt \epsilon\text{,}\) no matter how small \(\epsilon\) is, by making the distance from \((x,y)\) to \((a,b)\) 鈥渟mall enough鈥�.

Recall that a function \(f(x)\) is continuous at \(x=a\) if \(\ds\lim_{x\to a}f(x)=f(a)\text{.}\) We can say exactly the same thing about a function of two variables: \(f(x,y)\) is continuous at \((a,b)\) if \(\ds\lim_{(x,y)\to (a,b)}f(x,y)=f(a,b)\text{.}\)

The function \(f(x,y)=3x^2y/(x^2+y^2)\) is not continuous at \((0,0)\text{,}\) because \(f(0,0)\) is not defined. However, we know that \(\ds \lim_{(x,y)\to(0,0)}f(x,y)=0\text{,}\) so we can make a continuous function, by extending the definition of \(f\) so that \(f(0,0)=0\text{.}\) This surface is shown in 贵颈驳耻谤别听7.2.

Note that we cannot extend the definition of the function in 贰虫补尘辫濒别听7.11 to create a continuous function, since the limit does not exist as we approach \((0,0)\text{.}\)

Fortunately, the functions we will be working with will usually be continuous almost everywhere. As with single variable functions, two classes of common functions are particularly useful and easy to describe. A polynomial in two variables is a sum of terms of the form \(ax^my^n\text{,}\) where \(a\) is a real number and \(m\) and \(n\) are non-negative integers. A rational function is a quotient of polynomials.