Advanced Calculus fi..

Advanced Calculus, Fifth EditionTHEOREM 55 (M-Test for Integrals) Let M(t) be continuous for a 5 t < w;let f (t, x) be continuous in t for a 5 t < oo for each x of a set E. Iffor x in E andI f (t, x)l 5 M(t)converges, thenis uniformly and absolutely convergent for x in E.THEOREM 56 If f (t, x) is continuous in t and x for a 5 t < oo, c 5 x 5 d,andis uniformly convergent for c 5 x 5 d, then the function F(x) defined by thisintegral is continuous for c 5 x 5 d.THEOREM 57 If f (t, x) is continuous in t and x for a 5 t < w, c 5 x 5 d,andis uniformly convergent to F(x) for c 5 x 5 d, thenld F(x) dx = lw ld f (t, x) dx dl.THEOREM 58 If f (t, x) is continuous in t and x and has a derivative a flax,which is continuous in t and x for a 5 t < oo and c 5 x 5 d, and the integralsconverge, the second one uniformly, for c 5 x 5 d, thenhas a continuous derivative for c 5 x 5 d, andF1(x) = lmz(t , x) dt.These theorems are proved exactly as for series.\.i. - -,-*+, a