My problem is that I have a function: $f\colon\mathbb R\to\mathbb R$ with the property that $f(kx) = kf(x)$ for all $k,x \in\mathbb R$.
a) I shall show that $f(0)=0$
b) If $f$ is not the zero-function so that there is at least one $a\in\mathbb R$ with $f(a) \ne 0$, then $\forall x \in\mathbb R\colon f(x) = 0 \Leftrightarrow x=0$.
I don't really know how I shall proceed... Is it enough to say if: $k=0\Rightarrow kf(x)=0\cdot f(x)=0$ with $kf(x) = f(kx) = 0$ ? and how can I prove b) ?