Question

已知函数f(x)对一切x.y属于R,都有f(x+y)=f(x)+f(y).

求证:f(x)是奇函数.(60分中内回答)

Answer 1

设 x=y=0

则：
f(0+0)=f(0)+f(0)
即
f(0)=0

设y=-x

则
f(x+(-x)) = f(x) + f(-x)
即
f(0) = f(x) + f(-x)

又 f(0) = 0

则
f(-x) = -f(x)

Answer 2

x = y = 0时
f(0)=f(0) + f(0)
y = -x时
f(x)+f(-x) = f(0);
所以
f(x) = -f(-x);