A boundary point of the immediate basin of attraction of an attracting fixed point for a continuous function is either a non-attracting periodic point of period $2$ or a non-attracting fixed point. If $r \in [1,3]$ there are no real points of period $2$ and the only other fixed point is $0$. Thus all $a \in (0,\infty)$ are attracted to the attracting fixed point $(r-1)/r$.

A boundary point of the immediate basin of attraction of an attracting fixed point for a continuous function is either a non-attracting periodic point of period $2$, or a non-attracting fixed point, or a point mapped to a non-attracting fixed point. If $r \in [1,3]$ there are no real points of period $2$, the only other fixed point is $0$, and the only other point mapped to $0$ is $1$. Thus all $a \in (0,1)$ are attracted to the attracting fixed point $(r-1)/r$.