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Arkansas Tech UniversityMATH 2934: Calculus III

Dr. Marcel B. Finan

6 Limits and Continuity of Functions of TwoVariables

Let h denote the elevation of the terrain above sea level. Then h can beconsidered as a function of longitude and altitude. If we pick a point on aflat terrain we notice that the values of nearby points are close to the valueof the point. We say that h is continuous at that point. On the other hand,there are places on the earths surface where the elevation changes abruptly.We give a special name to such places: cliffs. Near either the base or thetop of a cliff the terrain may be fairly level and smooth. However as we

approach the cliff, there is a large and abrupt change in elevation. Within afew feet the terrain elevation may change by hundreds of feet (either upwardsor downwards). This break or sudden change in the ground elevation canbe considered a discontinuity.The above example illustrates the ideas of continuity and discontinuity. Roughlyspeaking, a function is said to be continuous at a point if its values at placesnear the point are close to the value at the point. If this is not the case thenwe say that the function is discontinuous.In this section, we present a formal discussion of the concept of continuity offunctions of two variables. Our discussion is not limited to functions of two

variables, that is, our results extend to functions of three or more variables.The definition of continuity requires a discussion of the concept of limits.Before we introduce this concept for functions in two variables, let us recallthe one dimensional version:Let f be a function and a be a point in its domain. We say that f(x) hasa limit L at a if and only if for every > 0 there exists a positive number depending on such that for any x in the domain of f with the property0 < |x a| < we have |f(x) L| < . In symbol, we write

limxa

f(x) = L

or f(x) L as x a.Geometrically, the definition says that for any > 0 (as small as we want),there is a > 0 (sufficiently small) such that any point inside the interval(a , a + ) is mapped to a point inside the interval (L , L + ) as shownin the figure below.

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A similar definition extends to functions in two variables: We say that L is

the limit of a function f at the point (a, b), writtenlim

(x,y)(a,b)f(x, y) = L

if f(x, y) is as close to L as we please whenever the distance from the point(x, y) to the point (a, b) is sufficiently small, but not zero.Using definition we say that L is the limit of f(x, y) as (x, y) approaches(a, b) if and only if for every given > 0 we can find a > 0 such that for anypoint (x, y) where 0 0, we can find an open punc-tured disk (i.e. without the center and the boundary) centered at (a, b) suchthat for any point (x, y) inside the disk the difference f(x, y) L is within, i.e., L < f(x, y) < L + . Figure 5.1 illustrates this.

Figure 5.1

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Example 6.1

Let f(x, y) = x2

+ y2

. Is lim(x,y)(1,1) f(x, y) = 3?

Solution.

Let = 0.1. Is there a > 0 such that all the points (x, y) inside theopen disk with radius and centered at (1, 1) satisfy 2.9 < f(x, y) < 3.1?Clearly, any such open disk will share points with the open disk centeredat (1, 1) and with radius 0.2. But any point (x, y) in this latter disk sat-isfies (x 1)2 + (y 1)2 < 0.04 or x2 + y2 2(x + y) + 2 < 0.04. Since0.8 < x < 1.2 and 0.8 < y < 1.2 we find 3.2 < 2(x + y) < 4.8. This impliesthat f(x, y) = x2 + y2 < 0.04 2 + 4.8 = 2.84 < 2.9. Hence any point in thedisk centered at (1, 1) and radius 0.2 will fall outside the interval (2.9, 3.1).

We conclude that lim(x,y)(1,1) f(x, y) = 3

As in the case of functions of one variable, limits of functions of two variablespossess the following properties: The limit, if it exists, is unique. The limit of a sum, difference, product, is the sum, difference, product oflimits. The limit of a quotient is the quotient of limits provided that the limit inthe denominator is not zero.

We can now define what we mean by continuity in terms of limit. Intuitively,we expect our definition to support the idea that there are no breaks orgaps in the function if it is continuous. The continuity of functions of twovariables is defined in the same way as for functions of one variable:A function f(x, y) is continuous at the point (a, b) if the following two con-ditions are satisfied:(a) f(a, b) exists;(b) lim(x,y)(a,b) f(x, y) = f(a, b).

A function is continuous on a region R in the xyplane if it is con-tinuous at each point in R. A function that is not continuous at (a, b) is said

to be discontinuous at (a, b).Since the condition lim(x,y)(a,b) f(x, y) = f(a, b) means that f(x, y) is closeto f(a, b) when (x, y) is close to (a, b) we see that our definition does indeedcorrespond to our intuitive notion that the graph of f(x, y) has no gaps orbreaks around (a, b).

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Like the case of functions of one variable, it can be shown that sums, prod-

ucts, quotients (where denominator function is not zero), and compositionsof continuous functions are also continuous.

Example 6.2

Show that

f(x, y) =

x2y

x2+y2(x, y) = (0, 0)

0 (x, y) = (0, 0)

is continuous at (0, 0).

Solution.

This function is clearly continuous everywhere except at (possibly) (0, 0).Lets check continuity at (0, 0). Let > 0 be given. Can we find a > 0 suchthat if 0