Advanced Calculus fi..

80 Advanced Calculus, Fifth EditionNow one verifies easily that the particular F[u, v] r u + v is continuous for allvalues of 14 and v. If (2.10) is applied to this choice of F, one findslim [f(x. y) + ~ (x. ?.)I = UI + VI..r + .r,y+y,which is (2.7); by the same reasoning, one concludes that if f (.r, y) and g(x, y) arecontinuous at (.rl. vl), then so is f (x. y) + g(.r, y).-The statements about products and quotients follow in the same way by considerationof the special functions F r u . v and F ulv. One need only showthat these functions are continuous (for v # 0 in the second case). This can be donedirectly by applying the preceding c. S definition or as follows. One shows that thefunctions u + v and u - v are continuous functions of u and v and also that awand w2 are continuous functions of w (theorems on functions of one variable). Itfollows from the function-of-function rule just proved that (u + v)' and (u - v)' arecontinuous and hence thatis continuous for all (14, v) and we-obtain the conclusions about the product f . g.Next one shows that I /v is a continuous function of v for v # 0 (function ofone variable) and hence that u/v u . (Ilv) is a continuous function of u and vfor v # 0. If, for example, vl > 0, then we take Do to be the half-plane v > 0 in theuv-plane. From (2.6) we conclude that g(.r, y) > v1 /2 > 0 in some neighborhood Dlof radius 6 of (.rI,?I)(except possibly at (XI, y,) itself). In the discussion of limitsandcontinuity, (.r. v ) can be restricted to Dl, so that the values (u, v) = ( f (x, y ), g(x, y))lie in Do, where F = u/v is continuous. Hence the previous reasoning applies tou/v.The theorem above can be restated in analogous form for functions of threeor more variables and, in the case of the composite function, for combinations offunctions of one and two variables, one and three variables, and so on:F[f (-r, y)I,F[f (f). g(t)l, F[ f (.r, ?I. :)I. F f t t ) ( t ] . ...By virtue of this theorem, one can conclude that polynomial functions such asare continuous for all values of the variables, whereas rational functions such asx'y - .rw = l-x'-yare continuous except where the denominator is 0.Mappings,from V" to Vm. At times, one considers sets of functions of n variables:?.I = XI.. . . ,.T,,).ym = trn(ll, . . .. X,,).(2.12)