※ 引述《HUJIKOLP (UUOJKLJLKJK)》之铭言： : Let f be continuous on a rectangular region (a,b)X(c,d) : prove that the function F defined by : F(X)=Sf(x,y)dy 积c~d x属於(a,b)is continuous on(a,b) 有错请指正罗~~ d F(x)=∫ f(x,y)dy c d F(x+h)=∫ f(x+h,y)dy c d d lim F(x+h)-F(x) = lim [(∫ f(x+h,y)dy) -(∫ f(x,y)dy)] h→0 h→0 c c d =lim ∫[f(x+h,y)-f(x,y)]dy fix x, and let f(x+h,y)-f(x,y)=G(y) h→0 c d =lim ∫ G(y)dy h→0 c (let P={c=a1,a2,....an=d} is a partition of [c,d] Max|P|→0 as n→∞) 1 =lim lim --- ΣG(ai) h→0 n→0 n 1 =lim --- Σ lim G(ai) n→0 n h→0 d = ∫ {lim [f(x+h,y)-f(x,y)]} dy c h→0 ----------------------- 这项因为f在(a,b)X(c,d)连续故为0 =0 lim F(x+h) = F(x) h→0 : ＝≒≠≡≡＜＞≦≧ : ∴∵∫∮∩∪㏑㏒√→∞∠△° : ⅠⅡⅢⅣⅤⅥⅦⅧⅨⅩ : ΑΒΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡΣΤΥΦΧΨΩ : αβγδεζηθικλμνξοπρστυφχψω --

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