Part 3: Definitions and Theorems

If we use the notation x = á x,y ñ and p = á p,q ñ , then

lim ( x,y) ® ( p,q)   f( x,y) = L  is  the  same  as   lim x® p  f( x) = L

Moreover, the use of vectors shows the similarity of the definition of the limit in two variables to the definition of the limit in one variable.  It also allows us to state definitions and theorems in their most general form:

To begin with, let us define any connected open set that contains a point p to be a neighborhood of p.  For example, an open ball of radius d > 0 about p, which is the set of all x such that || x-p || < d, is a neighborhood of p.  Any open rectangle containing p is also a neighborhood of p.  

The neighborhood concept allows us to state a general definition of the limit.       

Definition 2.2: The limit

lim x® p  f( x) = L

means that for all e > 0, there is an open neighborhood O of p such that if x is in O and if x ¹ p, then

| f( x) -L| < e

Definition 2.2 can be restarted in terms of open balls as follows:       

Definition 2.3 (Open Balls as Neighborhoods): The limit

lim x® p  f( x) = L

means that for all e > 0, there is a d > 0 such that

if  0 < | | x-p| | < d,  then  | f( x) -L| < e

       

The similarity of definition 2.3 to the single-variable definition of the limit is due to the use of vector notation.  The vector notation allows a straightforward generalization of other concepts in calculus as well.       

Definition 2.3: A function f( x) is continuous at a point p if f( p) is defined and

lim x® p  f( x) = f( p)

       

The following theorem is similarly a direct consequence of definition 2.3.        

Theorem 2.4: If f and g are continuous at a point (p,q) and k is a number, then kf, f-g, f+g, and f·g are also continuous at ( p,q) . Moreover, if g(p,q) ¹ 0, then f/g is also continuous at ( p,q) .

       

As a result, if f( x,y) is an arithmetic combination of functions which are continuous at ( p,q) , then f(x,y) is itself continuous at ( p,q) . For example, f( x,y) = cos( x) sin( y) is continuous everywhere since cos( x) and sin(y) are continuous for all x and y, respectively.

In addition, if ( p,q) is a boundary point of an open region on which f( x,y) is continuous, if f( x,q) is continuous as a function of x on a closed interval with p as an endpoint, and if f( p,y) is continuous as a function of y on a closed interval with q as an endpoint, then we say that f(x,y) is also continuous at the point ( p,q) .       

EXAMPLE 4    Where is the function f( x,y) = x^1/2+y^1/2 continuous?       

Solution: The function x^1/2 is continuous when x ³ 0, and the function y^1/2 is continuous when y ³ 0. Thus, f(x,y) is continuous on its entire domain, which is

dom( f) = { ( x,y)     |  x ³ 0    and   y ³ 0}