Question 134
Consider the function $f:(0, \infty) \rightarrow \mathbf{R}$ with the property that
$$
f(x y)=f(x)+f(y) \text{ for all } x, y \in (0, \infty)
$$
a) Show that $f(1)=0$ and $f\left(\frac{1}{x}\right)=-f(x)$ for all $x \in (0, \infty)$;
b) Show that if $1$ is the only solution of the equation $f(x)=0$, then the function $f$ is injective.
$$
f(x y)=f(x)+f(y) \text{ for all } x, y \in (0, \infty)
$$
a) Show that $f(1)=0$ and $f\left(\frac{1}{x}\right)=-f(x)$ for all $x \in (0, \infty)$;
b) Show that if $1$ is the only solution of the equation $f(x)=0$, then the function $f$ is injective.