4.2 Limits and continuity

We have now examined functions of more than one variable and seen how to graph them. In this section, we see how to take the limit of a function of more than one variable, and what it means for a function of more than one variable to be continuous at a point in its domain. It turns out these concepts have aspects that just don’t occur with functions of one variable.

Limit of a function of two variables

Recall from Section $2.2$ the definition of a limit of a function of one variable:

Let $f\left(x\right)$ be defined for all $x\ne a$ in an open interval containing $a.$ Let $L$ be a real number. Then

if for every $\epsilon >0,$ there exists a $\delta >0,$ such that if $0<\left|x-a\right|<\delta$ for all $x$ in the domain of $f,$ then

Before we can adapt this definition to define a limit of a function of two variables, we first need to see how to extend the idea of an open interval in one variable to an open interval in two variables.

The idea of a $\delta$ disk appears in the definition of the limit of a function of two variables. If $\delta$ is small, then all the points $\left(x,y\right)$ in the $\delta$ disk are close to $\left(a,b\right).$ This is completely analogous to $x$ being close to $a$ in the definition of a limit of a function of one variable. In one dimension, we express this restriction as

In more than one dimension, we use a $\delta$ disk.

Proving that a limit exists using the definition of a limit of a function of two variables can be challenging. Instead, we use the following theorem, which gives us shortcuts to finding limits. The formulas in this theorem are an extension of the formulas in the limit laws theorem in The Limit Laws .