I'm a beginner in differential topology and was studying: Introduction to Smooth Manifolds by John M. Lee. And came across an example (Example 2.13 e ) :
Could someone please explain how one could deduce that the quotient map, used to define the projective space, is smooth based on the above example.
Additionally here is the Example 1.33:
In both examples, the maps are smooth simply by definition: map $f\colon (x_1,\ldots,x_n) \mapsto (f_1(x_1,\ldots,x_n), \ldots, f_m(x_1, \ldots,x_n))$ is called smooth if for all possible $j, k$ there exists derivative $\frac{\partial f_j}{\partial x_k}$, which is a continuous function in $x_1, x_2, \ldots, x_n$. In the first example, $\frac{\partial}{\partial x_j}(\frac{x_j}{x_i})=\frac{1}{x_i}$ and $\frac{\partial}{\partial x_i}(\frac{x_j}{x_i})=-\frac{x_j}{x_i^2}$ which are both continuous functions. Similarly for example $1.33$.
Edit. The definition given above is actually the definition of a $C^1$-smooth map. But I did not take into account that the word 'smooth' commonly means '$C^{\infty}$-smooth', i. e. there exist all possible higher partial derivatives $\frac{\partial^N f}{\partial x_{j_1}\partial x_{j_2} \ldots \partial x_{j_N}}$ (which are also automatically continuous). It is easy to see that in your example this condition holds: by differentiation of $\frac{x_j}{x_i}$ one can obtain only rational functions with some power of $x_i$ in the denominator. Such functions are defined and differentiable if $x_i \ne 0$.