Consider the following definition, where immersions are defined:
Later the author states:
The problem: In the definition, a regular function is defined to have constant rank in particular. But later it is stated, as if it were something that would need a proof, that the rank of an immersion (which in particular is a regular function), is constant in a neighborhood. This doesn't make sense.
What am I missing here? Maybe the author messed up the definition of "regular function" and assumed too much?
Later the author states "Let $\varphi$ be regular at $0$, i.e. $D\varphi (0)$ is injective. How can I make sense of this, if regularity was defined as a property holding for the whole domain, not just a single point? (Also, $D\varphi (0)$ being injective is something that pertains to an immersion, not a regular function, according to the definitions above.)