I want to show that 2 maximal atlases on $\mathbb{R}$ are distinct. Define $$ \phi_r(t)=\begin{cases} t & t\leq0 \\ rt & t\geq0 \end{cases} $$
Now make two maximal atlases; one from $(\mathbb{R},\phi_r)$, and the other from $(\mathbb{R},\phi_s)$ where we have that $0<r<s$.
I know that on the real line as we usually have it, neither of $\phi_r,\phi_s$ would be smooth, but because we make a differentiable structure from them they now count as being smooth. But how can I show that they are incompatible?
From my understanding, if we compose one of them with the inverse of the other we will get for example $$\phi_s\circ\phi_r^{-1}(\phi_r(\mathbb{R}))=\begin{cases} t & t\leq0 \\ \frac{s}{r}t & t\geq0. \end{cases}$$ as a mapping from $\phi_r(\mathbb{R})\rightarrow \mathbb{R}$.
Now this shouldn't be smooth, but I don't understand how I can reason that that is the case because we built the differentiable structures from these two maps. I think I am just slightly lost with some of the detailed specifics of what I have to show. Would anybody be able to clarify the nuance that I am missing?