Let $f$ a function defined on $]0,+\infty[$ and checking : $$\forall x,y>0,\ f(xy)=f(x)+f(y)$$ Assume that $f$ is bounded on $]1-\eta,1+\eta[$($\subset]0,+\infty[$). Let $\alpha =\underset{x\in ]1,1+\eta[}{\inf}f(x)$ and $\beta =\underset{x\in ]1,1+\eta[}{\sup}f(x)$. So we have :
- If $\alpha >0$, $f$ is increasing on $]0,+\infty[$2)
- If $\beta <0$, $f$ is decreasing on $]0,+\infty[$
- And if $\alpha <0<\beta$, how falling into a contradiction in this case ???!!!!