Question

若函数f(x)对任意的x、y∈R满足f(x+y)=f(x)+f(y)

则下列各式中不正确的是
A、f(0)=0
B、f(3)=3f(1)
C、f（1/2）=1/2f(1)
D、对于任意的x∈都有F(-x)f(x)＜0
答案选D。请说明理由

Answer 1

令x=0， y=0
那么
f(x + y) = f(x) + f(y)
得到
f(0+0) = f(0) + f(0) = f(0)
也就是
2f(0) = f(0)
所以f(0)=0
A正确

f(3) = f(2 + 1) = f(2) + f(1) = f(1+1)+f(1) = f(1) + f(1)+ f(1)= 3f(1)
B正确

f(1) = f(1/2) + f(1/2) = 2f(1/2)
两边除以2，得到
f(1/2) = 1/2 f(1)
C正确
当x = 0时
f(-x)=f(0) = 0
f(x) = f(0) = 0
所以
f(-x)f(x) = 0*0 = 0
不满足不等式
所以D错误
选D