2.2 Limits and continuity In order to introduce the concepts of limit and continuity for functions of more than one variable we need first to generalise the concept of neighbourhood of a point from R to R 2 and R 3. Let us start by recalling the corresponding definition for functions of one variable. 2.2.1 Neighbourhood of a point in one, two and three dimensions Definition: If x 0 is a point in the real line and ε > 0, then the set of points x in the real line whose distances from x 0 are less than ε is called an open interval about x 0 or a neighbourhood of x 0. If we denote this neighbourhood by N ε (x 0 ), then N ε (x 0 ) = {x : x x 0 < ε} (2.9) Definition: If (x 0, y 0 ) is a point in the 2-dimensional xy-plane and ε > 0, then the set of points (x, y) in the plane whose distances from (x 0, y 0 ) are less than ε is called an open disk of radius ε about (x 0, y 0 ) or, more simply, a neighbourhood of (x 0, y 0 ). If we denote this neighbourhood by N ε ((x 0, y 0 )), then { N ε ((x 0, y 0 )) = (x, y) : } (x x 0 ) 2 + (y y 0 ) 2 < ε. (2.10) Note: Since the equation (x x 0 ) 2 + (y y 0 ) 2 = ε 2, (2.11) defines a circle of radius ε centered at the point (x 0, y 0 ), the neighbourhood of (x 0, y 0 ) are all the points contained inside that circle (that is, a disk) but not the points on the circle. Figure 3: Neighbourhood of a point in one, two and three dimensions. The regions D are arbitrary subspaces of R 2 or R 3 where the point and its neighbourhood live. Definition: If (x 0, y 0, z 0 ) is a point in the 3-dimensional xyz-plane and ε > 0, then the set of points (x, y, z) in the plane whose distances from (x 0, y 0, z 0 ) are less than ε is called an open ball of radius ε about (x 0, y 0, z 0 ) or, more simply, a neighbourhood of (x 0, y 0, z 0 ). If we denote this neighbourhood by N ε ((x 0, y 0, z 0 )), then { N ε ((x 0, y 0, z 0 )) = (x, y, z) : } (x x 0 ) 2 + (y y 0 ) 2 + (z z 0 ) 2 < ε. (2.12) 8
Note: Since the equation (x x 0 ) 2 + (y y 0 ) 2 + (z z 0 ) 2 = ε 2, (2.13) defines a sphere of radius ε centered at the point (x 0, y 0, z 0 ), the neighbourhood of (x 0, y 0, z 0 ) are all the points contained inside that sphere (that is, a ball) but not the points lying on the sphere. It is convenient here to introduce some notions related to sets in the 2-dimensional space: Definition: Let S be a set of points in the plane: We say that a point (x 0, y 0 ) is a boundary point of S if every neighbourhood of (x 0, y 0 ) contains at least one point in S and at least one point not in S. We say that S is closed (or is a closed set) if every boundary point of S belongs to S. We say that S is open (or is an open set) if no boundary point of S belongs to S. The interior of S is the set of all points in S that are not boundary points of S. The exterior of S is the set of all points that are not in S and not boundary points of S. Figure 3: Boundary points, interior and exterior points corresponding to the set S defined by x 2 +y 2 1, that is the closed disk of radius 1. 2.2.2 Limits of functions of two variables The concept of limit of a function of several variables is similar to that for functions of one variable. For clarity we will present here only the case of functions of 2 variables; the general case is a natural generalization of that one. 9
Definition: We say that a function f(x, y) approaches the limit L as the point (x, y) approaches the point (x 0, y 0 ) and we write lim f(x, y) = L, (2.14) if all points of a neighbourhood of (x 0, y 0 ), except possibly the point (x 0, y 0 ) itself, belong to the domain D(f) of f, and if f(x, y) approaches L as (x, y) approaches (x 0, y 0 ). In other words, the closer the point (x, y) is to (x 0, y 0 ), the closer is the function f(x, y) to the limiting value L. A more formal definition of limit: We say that lim f(x, y) = L, (2.15) every neighbourhood of (x 0, y 0 ) contains points (x, y) in D(f), and if and only if for every number ε > 0 there exists another number δ(ε) > 0 such that whenever (x, y) D(f). if 0 < (x x 0 ) 2 + (y y 0 ) 2 < δ(ε) then f(x, y) L < ε, (2.16) Laws of limits: All usual laws of limits extend to functions of several variables in the obvious way. For example if then Existence of the limit: If a limit exists it is unique. lim f(x, y) = L and lim g(x, y) = M, (2.17) lim (f(x, y) ± g(x, y)) = L ± M (2.18) lim f(x, y)g(x, y) = LM, (2.19) f(x, y) lim g(x, y) = L M. (2.20) For a function of one variable f(x) the existence of the limit lim x x0 f(x) implies that the function f(x) approaches the same finite number as x approaches x 0 from either the right or the left. For a function of 2 variables lim (x,y) (x0,y 0 ) f(x, y) = L exists only if f(x, y) approaches the same number L no matter how (x, y) approaches (x 0, y 0 ) in the xy-plane. That means that (x, y) can approach (x 0, y 0 ) along any curve in the xy-plane which goes through (x 0, y 0 ). Notice that it is not necessary that f(x 0, y 0 ) = L, even if f(x 0, y 0 ) is defined. Let us now exemplify these definitions with various examples: 10
Example 1: Let us consider the following limits lim 2x (x,y) (1,1) y2 = 2 1 = 1, (2.21) lim x2 y = x 2 0y 0, (2.22) lim y sin(xy) = 2 sin(2π) = 0. (2.23) (x,y) (π,2) All these examples show limits which exist, in the sense that the functions approach the same limit no matter how (x, y) approaches the limiting point. In fact, in all these cases the limit of the function coincides with the value of the function at the limiting point. As we said above this is not always the case. Example 2: Let and try to compute the limit f 2 (x, y) = 2xy x 2 + y 2, (2.24) lim f 2(x, y). (2.25) It is not difficult to prove that this limit does not exist. This is due to the fact that the function f(x, y) approaches different limiting values depending on the way the point (x, y) approaches (0, 0). To see that, suppose that we take the limit (x, y) (0, 0) along curves parameterized by y = kx, where k is an arbitrary real number different from 0, lim f 2(x, kx) = (x,kx) (0,0) lim (x,kx) (0,0) 2kx 2 x 2 + k 2 x 2 = 2k 1 + k 2. (2.26) Obviously, the value of the limit is different for each different value of k. Therefore the limit is not unique which implies it does not exist. This is also an example of a function which is not well defined at the limiting point (0, 0), since f(0, 0) = 0/0 = indeterminate. Example 3: Let We want to compute the limit f 3 (x, y) = 2x2 y x 4 + y 2. (2.27) lim f 3(x, y). (2.28) We can try to compute the limit along lines parameterized as before, y = kx with k 0. We obtain lim f 2kx 3 3(x, kx) = lim (x,kx) (0,0) x 0 x 4 + k 2 x 2 = lim 2kx x 0 x 2 = 0. (2.29) + k2 So the limit along any straight line y = kx is 0. We might be then tempted to conclude that lim f(x, y) = 0, but this would be incorrect. To see that let us now take the limit along curves of the form y = kx 2. Then lim f 3(x, kx 2 2kx 4 ) = lim (x,kx 3 ) (0,0) x 0 x 4 + k 2 x 4 = 2k 1 + k 2. (2.30) So, the limit along curves y = kx 2 is not 0 but gives a different value for each different choice of k. Therefore the limit (2.28) does not exist. 11
Example 4: Let and prove that the limit f 4 (x, y) = x2 y x 2 + y 2. (2.31) lim f 4(x, y) = 0. (2.32) We can try to prove that by choosing some particular curves along which to take the limit, as in the previous examples. However, to be really sure that the limit is always the same, no matter which curve we take, we can try to prove (2.32) from the rigorous definition of the limit given in (2.16). Since x 2 x 2 + y 2 and y 2 x 2 + y 2, we have that f 4 (x, y) 0 = x 2 y x 2 + y 2 y x 2 + y 2. (2.33) So, formally if we consider a number ε > 0 and we take δ(ε) = ε it is guaranteed that if then 0 < x 2 + y 2 < ε, (2.34) f 4 (x, y) 0 < ε, (2.35) therefore the limit according to definition (2.16) exists and is (2.32). Figure 4: Graphic representation of the function f 4 (x, y) = x 2 y/(x 2 + y 2 ). 2.2.3 Continuity of functions of two variables Definition: The function f(x, y) is continuous at the point (x 0, y 0 ) if lim f(x, y) = f(x 0, y 0 ). (2.36) 12
Note: Notice that it is not always true that the limit of a function at a certain point coincides with the value of the function at that point. Therefore the condition (2.36) is a non-trivial constraint. For example, we can define the following function f(x, y) = { 0 if (x, y) (0, 0) 1 if (x, y) = (0, 0) (2.37) It is clear that lim f(x, y) = 0, (2.38) since when performing the limit we approach the point (0, 0) infinitely but we never exactly reach it. Therefore the value of the limit is in this case the value of the function everywhere but at (0, 0), which is by definition 0. However f(0, 0) = 1. Therefore, this function is discontinuous at (0, 0). We can easily make this function continuous everywhere by simply changing its value at the origin to f(0, 0) = 0. 13