Limits and Continuity

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 and derivatives.

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:

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 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).