Limits and Continuity of Functions of Two or More Variables

Introduction

Recall that for a function of one variable, the mathematical statement

means that for x close enough to c, the difference between f(x) and L is "small". Very similar definitions exist for functions of two or more variables; however, as you can imagine, if we have a function of two or more independent variables, some complications can arise in the computation and interpretation of limits. Once we have a notion of limits of functions of two variables we can discuss concepts such as continuity andderivatives.

Limits

The following definition and results can be easily generalized to functions of more than two variables. Let f be a function of two variables that is defined in some circular region around (x_0,y_0). The limit of f as x approaches (x_0,y_0) equals L if and only if for every epsilon>0 there exists a delta>0 such that f satisfies

whenever the distance between (x,y) and (x_0,y_0) satisfies

We will of course use the natural notation

when the limit exists. The usual properties of limits hold for functions of two variables: If the following hypotheses hold:

and if c is any real number, then we have the results:

Linearity 1:
Linearity 2:
Products of functions:
Quotients of functions:

(provided L is non-zero)

The linearity and product results can of course be generalized to any finite number of functions:

The limit of a sum of functions is the sum of the limits of the functions.
The limit of a product of functions is the product of the limits of the functions.

It is important to remember that the limit of each individual function must exist before any of these results can be applied.

Example

Find the limit of the function f(x,y)=x^3+2yx^2 as (x,y) approaches (1,2). Since the limits of the functions x^3, x^2, and y all exist, we may apply the linearity and product properties of limits to get

Example

The product property of limits cannot be applied to the function f(x,y)=xlog(y) as (x,y) approaches (0,0) since the log function approaches minus infinity as y approaches zero. L'Hopital's rules must be used for this type of problem.

With functions of a single variable, if the limits of a function f as x approached a point c from the left and right directions differed, then the function was found to not have a limit at that point. The same is true for functions of two variables, but now there are an infinite number of directions to choose from rather than just two. Consider the function xy/(x^2+y^2). As (x,y) approaches (0,0) along the x-axis (y=0), the function has limit 0; but, as (x,y) approaches (0,0) along the line y=x, the function has limit 1/2. Thus, the function f does not have a limit as (x,y) approaches (0,0).

Continuity

A function f of two variables is continuous at a point (x_0,y_0) if

f(x_0,y_0) is defined
exists

This definition is a direct generalization of the concept of continuity of functions of one variable. The three requirements ensure that f does not oscillate wildly near the point, does not become infinite at the point, or have a jump discontinuity at the point. These are all familiar properties of continuous functions. As with functions of one variable, functions of two or more variables are continuous on an interval if they are continuous at each point in the interval.

Continuous functions of two variables satisfy all of the usual properties familiar from single variable calculus:

The sum of a finite number of continuous functions is a continuous function.
The product of a finite number of continuous functions is a continuous function.
The quotient of two continuous functions is a continuous function wherever the denominator is non-zero.

Example

The functions sin(xy), x^2y^3+ln(x+y), and exp(3xy) are all continuous functions on the xy-plane, whereas the function 1/xy is continuous everywhere except the point (0,0).

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