I attempt to understand the definition and examples of Lie group supplied by An Introduction to Manifolds by Loring Tu (Second Edition, page no. 66). The definition is given below.
The first example is as follows.
The Euclidean space $\mathbb{R}^n$ is a Lie group under addition.
For $\mathbb{R}^n$ to be a Lie group, the multiplication map $\mu: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n, \mu(x, y) = x + y$, and the inverse map $i: \mathbb{R}^n \to \mathbb{R}^n, i(x) = \frac{1}{x}$, need to be $C^{\infty}$. I am convinced that $\mu$ is a $C^{\infty}$ map on $\mathbb{R}^n \times \mathbb{R}^n$, but I don't see how the inverse map $i$ is $C^{\infty}$ at $x = 0 \in \mathbb{R}^n$. To understand it better, I considered the limiting case where $n = 1$ in $\mathbb{R}^n$. In that case the inverse map $i$ is not defined at $x = 0 \in \mathbb{R}$.
What am I missing here? How can then $\mathbb{R}^n$ be a Lie group under addition?
It doesn't make sense to talk about $\frac1x$ when $x\in\Bbb R^n$ and $n>1$. The inversion, in this context, is the map $x\mapsto-x$.