Advanced Calculus fi..

78 Advanced Calculus, Fifth Editionmore of the variables and thereby obtain level surfaces in 3-dimensional space orlevel curves in a plane.A function f (x, y) is said to be bounded when (x, y) is restricted to a set E,if there is a number M such that 1 f (x, y)/ < M when ( x, y) is in E. For example,z = x2 + y2 is bounded, with M = 2, if 1x1 < 1 and (y 1 < 1. The function z =< in.tan (x + y ) is not bounded for Jx + y 12.4 LIMITS AND CONTINUITYLet z = f (x, y) be given in a domain D, and let (xl . y l) be a point of D or a boundarypoint of D. Then the equationlim f(x. y) =Cx-x,v-Vlmeans the following: given any t > 0, a 6 > 0 can be found such that for every(x, y) in D and within the neighborhood of (xi, y,) of radius 6, except possibly for(xI, y ) itself, one hasIn other terms, if (x, y) is in D andthen (2.3) holds. Thus if the variable point (x. y) is sufficiently close to (but not at)its limiting position (x,, y,). the value of the function is as close as desired to itslimiting value c.If the point (xl, y,) is in D andthen f (x, y) is said to be continuous at ( XI, y,). If this holds for every point (x,, y,)of D, then f (x, y) is said to be continuous in D.The notions of limits and continuity can be extended to more complicated setsforexample, to closed regions. The preceding definitions can be repeated essentiallywithout change. Thus if f (x, y) is defined in a closed region R and (xl, yl) is in R.then (2.2) is said to hold if, for given t > 0, a 6 > 0 can be found such that (2.3)holds whenever (x, y) is in R and is within distance 6 of (x,, yl), but not at (x,, yl). If(2.5) holds, then f (x, y) is continuous at (xl, y,). Similar definitions hold iff (x, y)is defined only on a curve in the xy-plane. The notions of limits and continuity mustalways be considered in relation to the set in which the function is defined.Continuity for functions of two variables is a more subtle requirement than forfunctions of one variable. Such a simple function asis badly discontinuous at the origin, without becoming infinite there. For z has limit0 if (x, y) approaches the origin on the line x = y, has limit 1 if (x, y) approachesthe origin on the x axis, and has limit - 1 if (x, y) approaches the origin on the y axis.