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Consider the following definition, where immersions are defined:

enter image description here

Later the author states:

enter image description here

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.)

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The underlined part in second picture actually has nothing to do with the first picture. That is simply an application of the property of continuous functions.

Update

If not defined, then you may understand the regularity as

$f \colon U \to \mathbb R^n$ where $U \subseteq \mathbb R^k$ is regular at $x_0 \in U$ if $\mathrm Df(x_0)$ has full rank, i.e. $$\mathrm {rk}(\mathrm Df(x_0)) = \min \{n, k\}.$$

When the author says

i.e. $\mathrm Df(0)$ is injective

there should be some statements before about the dimensions of domain and codomain, otherwise this is not fully supported.

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  • $\begingroup$ But then it's nonsensical to formulate this remark by stating it as a property for immersions, right? (See please also my edit of the question.) $\endgroup$ – temo Aug 12 '18 at 15:13
  • $\begingroup$ @temo I just mean the reasoning of this sentence have not use these concepts. $\endgroup$ – xbh Aug 12 '18 at 15:17
  • $\begingroup$ So you agree it doesn't make sense to state that remark? $\endgroup$ – temo Aug 12 '18 at 15:20
  • $\begingroup$ @temo No. I do not know why the author give this remark, since I have not read this note. Maybe this is a review of multivariate calculus [if you are reading manifolds, i assume] or this would be useful for future reference. $\endgroup$ – xbh Aug 12 '18 at 15:23
  • $\begingroup$ If the definition of regularity at one point has not given before, then this might be a flaw of organizations of materials. $\endgroup$ – xbh Aug 12 '18 at 15:25

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