Advanced Calculus fi..

Chapter 2 Differential Calculus of Functions of Several Variables 79Thus no limiting value can be assigned at (0, 0). It should be noted that the levelcurves of this function are straight lines, all passing through (0, 0); this alone showsthat there is a discontinuity at the origin.However, the fundamental theorem on limits and continuity holds withoutchange:THEOREM Let u = f (x, y) and v = g(x. y) both be defined in the domain Dof the xy-plane. LetThenlim f'(.r, y) = ul. J@,g(x, y) = U I.I+\\v+y,Y-?If(x, Y) 11 Ilim = - (uI # 0)..r + x 1. y) tllIf f (.r, y) and g(x, y) are continuous at (xi. yl), then so also are the functionsprovided, in the last case. g(xl, yl) # 0.Let F(u. ti) be defined and continuous in a domain Do of the uv-plane and letF[f(.r, y),g(.r.y)] bedefinedfor(x, y)in D.Then, if(uI, vl)isin Do,If f(x, y) and g(.r. y) are continuous at (xl, yl). then so also is F[ f (x, y), g(x, y)].Proof. We first consider the composite function F[ f (x, y), g(x, y)], which is themost fundamental notion of the whole theorem. Since F[u, v] is assumed to becontinuous in Do, one haslim F[u. LI] = F[irl. tll];18 ' Ld II" L,l(2.1 1)by (2.6), as (.r , y) approaches (XI, y~ ). (u, v) approaches (u I, v,), so that by (2.1 l),lim F[f'(.r, y), g(x. y)] = F5,1@, f(x. y), lim g(x, y)] = F[ul, vl].X'.lI .r-.t 1v+ Vl v+ Yl v+ VlThus (2.10) is established. If f and g are continuous at (xI, y,), then f (xi, y,) = u Iand g(.r~, yl) = v,, so that by (2.10),.I-.XIy-r Vllim F[f(x. y). g(x. y)] = F[ f (XI. YI). XI. ?I)];that is, F[ f (s, y). g(x. y)] is continuous at (XI, yl).