Question

If $f(x+y)=f\left(x\right)+f\left(y\right)$. Then prove that $f\left(x\right)=x.f\left(1\right)$

Answer 1

$f(x+y)=f\left(x\right)+f\left(y\right)y=2:f\left(x\right)=f\left(x\right)+f\left(0\right)\Rightarrow f\left(0\right)=0(x=1y=1);f\left(2\right)=2f\left(1\right)(x=2y=1);f\left(3\right)=2f\left(1\right)+f\left(1\right)=3f\left(1\right)\Rightarrow f\left(x\right)=xf\left(1\right)$