I want to show $M=\mathbb{R}$ with an atlas of chart $U=\mathbb{R}$, $f:U\rightarrow\mathbb{R}$ such that $f(x)=2x$ when $x\geq 0$ and $f(x)=3x$ when $x\leq 0$. Then $M$ is a smooth manifold which is diffeomorphic to $\mathbb{R}$.
I got a part showing that $M$ is diffeomorphic to $\mathbb{R}$. Also $M$ is Hausdorff and second countable. However I am having trouble showing $M$ is a smooth manifold.