Suppose a function f is additive. Also $f(x)$ is positive whenever $x$ is positive and $f(x)$ is negative whenever $x$ is negative. My question is "Is it necessary that f is linear?" If yes then please give a proof and if no then I am looking for a counterexample.
I've heard that if an additive function is bounded, it must be linear. $f(x)\ge 0$ is a function bounded below. I search some previous asked questions here also in Google but couldn't find any satisfactory answer. Please help me. Thanks in advance
Yes, this is true. The thing to remember about additive functions on $\mathbb R$ is that they are automatically linear on the rational numbers due to the equations
$$f(nx)=nf(x)$$ $$f(\frac1n)=\frac1nf(x)$$
which follow from additivity. So $f(q)=qf(1)$ for rational $q$, and if we can use any kind of continuity argument we can fill in the gaps to show $f$ is linear on $\mathbb R$.
Here, your condition guarantees monotonicity since if $x>y$ we can write $x=y+a$ for positive $a$, and then $f(x)=f(y)+f(a)>f(y)$. Monotonicity allows us to fill in the gaps since $f(x)$ must be between $a_nf(1)$ and $b_nf(1)$ for any two rational sequences $a_n$ and $b_n$ which approach $x$ respectively from below and from above, so $f(x)=xf(1)$.