证(1)令y=∫(x)+g(x,则 Δy[f(x+△x)+g(x+△x)-[f(x)+g(x) △ x+A)-f(x)+18(x+A)-g(x △ im今=imf(x+Ax)-f(x) △x→0△x△x→0 △ lim8(x+△x)-g(x) f∫'(x)+g(x) Ax→0 △ 即[f(x)+g(x)=∫(x)+g(x) 同理可证[f(x)-g(x)=f(x)-g(x)2 (1). ( ) ( ), [ ( ) ( )] [ ( ) ( )] y f x g x y f x x g x x f x g x x x = +  +  + +  − + =   证 令 则 [ ( ) ( )] [ ( ) ( )] f x x f x g x x g x x x +  − +  − = +   0 0 0 ( ) ( ) lim lim ( ) ( ) lim ( ) ( ) x x x y f x x f x x x g x x g x f x g x x  →  →  →  +  − =   +  − + = +    即 [ ( ) ( )] ( ) ( ). f x g x f x g x + = +    同理可证 [ ( ) ( )] ( ) ( ). f x g x f x g x − = −   