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Can someone share some good scholarly references where the left and right inverse of functions are defined, and their properties analysed?

Even better if they are references that also discuss various different definitions in circulation, their advantages and drawbacks, and their different usefulness in different branches of mathematics.

pglpm
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  • See Left and right inverses – Mauro ALLEGRANZA Feb 02 '21 at 11:34
  • @MauroALLEGRANZA The wikipedia section on left/right inverses doesn't have a single reference (as of today). I don't see why I should believe what's written there. – pglpm Feb 02 '21 at 11:36
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    Have you tried with some textbook? Vladimir Zorich, Mathematical Analysis I (Springer, 2nd ed 2016) – Mauro ALLEGRANZA Feb 02 '21 at 11:39
  • @MauroALLEGRANZA Thank you! that's a good text. I'd like to find something that expands a little more on this topic, rather than leaving it to an exercise. – pglpm Feb 02 '21 at 11:44
  • @MauroALLEGRANZA Quite interesting that Zorich defines two functions to be equal if they have the same domain and range – even if their codomains are different. – pglpm Feb 02 '21 at 11:47
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    @pglpm That's in line with the traditional, and very reasonable, assumption that two functions should be equal if and only if $f(x)=g(x)$ for all $x$. To me this makes sense: if, on the contrary, you want them to be equal as morphism in the category of sets, then you should reference the same codomain as well, but I would think that in that case referencing the additional categorical machinery is somewhat necessary anyways. –  Feb 02 '21 at 11:52
  • @Gae.S. That's exactly what I'm looking for: a book that compares different conventions and explains their advantages and drawbacks, and why they end up being different in different branches of mathematics. I imagine (I don't know) that a topologist would find it more convenient to ensure the equality of codomains, for example. – pglpm Feb 02 '21 at 11:54
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    This google search produced several items of possible interest to you, such as: item 1 AND item 2 AND item 3 (a Ph.D. thesis; has lots of references) AND item 4 (another Ph.D. dissertation) – Dave L. Renfro Feb 02 '21 at 16:13
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    Also of possible interest are some of the references I give in this answer, especially the paper by Salomon Bochner (because it focuses more on modern ideas than the other papers). – Dave L. Renfro Feb 02 '21 at 16:26
  • Great references, @DaveL.Renfro Please feel free to post your first comment as an answer and also link your other answer. – pglpm Feb 02 '21 at 16:32
  • @DaveL.Renfro The keyword "mathematics teacher" made a world of difference. I hadn't thought of that. – pglpm Feb 02 '21 at 16:33
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    I was intending to search papers published in the journal The Mathematics Teacher, and after I quickly did the search (with quotes for some words), I noticed I wasn't getting anything from that journal, that I could see, and only then (!!) did I realize that I'd forgotten that "Mathematics Teacher" is a common phrase (all this took about 10 seconds). However, the first few hits I clicked on were so appropriate that I abandoned my original search idea and just used that first attempt. (I'll post an answer shortly --- busy right now.) – Dave L. Renfro Feb 02 '21 at 17:04
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    (update) After I finished some stuff I was working on (off-line) and came back here to post an answer, I began having a lot of problems with internet connection (connection kept getting lost), and I finally gave up trying. Checking things now, a few hours later, my internet connection seems fine, but since I'm busy with some other stuff, I'll take care of this tomorrow. – Dave L. Renfro Feb 02 '21 at 21:48

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The google search "mathematics teacher" "function" "codomain" produces several items of possible interest to you, such as:

The mistakes that are made by students with regard to functions: evidence from Erzincan by Kani Başibüyük, et al. (2016 math education paper; 10 pages)

Undergraduate mathematics students’ understanding of the concept of function by Caroline Bardini, et al. (2014 math education paper; 23 pages)

The Function Concept and University Mathematics Teaching by Olov Viirman (2014 Ph.D. dissertation; 99 pages)

Teaching Students to Communicate with the Precise Language of Mathematics: A Focus on the Concept of Function in Calculus Courses by Derrick S. Harkness (2020 Ph.D. dissertation; xii + 137 pages)

Also of possible interest are some of the references I give in this answer, especially The rise of functions by Salomon Bochner as it focuses more on modern ideas than the other papers.

Dave L. Renfro
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