Other names for the domain and range-straightening theorems?

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In summary, the conversation discusses the search for the wiki page on the domain-straightening theorem and the range-straightening theorem, also known as the canonical forms for submersion and immersion. The search did not produce any results, except for a page on straightening theorem for vector fields, which was not exactly what the speakers were looking for. They also mention a possible resource for the theorems in a real analysis context, such as a lecture note from MIT.
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I'm trying to find the wiki page (if it exists) on the domain-straightening theorem and the range-straightening theorem but searches for those words produce no results.

I've also seen those theorems by the name of "canonical forms for submersion and immersion" (in Gallot-Lafontaine-Hullin), which produces no result either.
 
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Bumping this thread, because I am in the same situation as the OP, and since I also haven't found anything. Well, I guess the one thing I found was http://en.wikipedia.org/wiki/Straightening_theorem_for_vector_fields, but that wasn't quite what I was looking for. I namely stumbled upon this theorem in real analysis, and while it seems similar to what the wikipedia page has there, it's still not the same.
 

1. What are the other names for the domain and range-straightening theorems?

The domain and range-straightening theorems are also commonly referred to as the monotonicity theorem or the increasing/decreasing function theorem.

2. What do the domain and range-straightening theorems state?

These theorems state that a function is either strictly increasing or strictly decreasing over its entire domain if its derivative is either always positive or always negative, respectively.

3. How are the domain and range-straightening theorems used in mathematics?

These theorems are used to analyze the behavior of functions and to prove the existence of extrema (maximum or minimum values) for a given function.

4. Are there any exceptions to the domain and range-straightening theorems?

Yes, there are exceptions when the function is defined on a non-continuous domain or when the function has a vertical asymptote. In these cases, the theorems may not apply.

5. Can the domain and range-straightening theorems be applied to all types of functions?

No, these theorems are specifically applicable to continuous functions. They may not hold true for discontinuous or piecewise-defined functions.

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