Consider the map $f: \mathbb{R}P^{2} \rightarrow \mathbb{R}^{3}$ given by $f([a,b,c])=(bc,ac,ab)$ where $\mathbb{R}P^{2}$ denotes the projective plane.
In order to show that $f$ is not an immersion in general, why it suffices to show that the map:
$g: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ given by $g(a,b,c)=(bc,ac,ab)$ is not an immersion in all $\mathbb{S}^{2}$?
I can see that if $g$ is not an immersion at a point then $f$ is not an immersion at the image of that point.
But why is the converse true? how do we know that if $f$ is not an immersion then $g$ is not an immersion? I mean how do we know that by checking those points where $g$ is not an immersion then we know there cannot be other points in $\mathbb{R}P^{2}$ for which $f$ is an immersion?