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This question was probably asked many times, but i couldn't find by searching. And i figure that i'll get an answer faster then by searching.

We we say :

functions on M

Does that mean that the function maps from $M$ to $M$, that it's image lies in $M$, or that it's arguments are in $M$ or is it very context dependent ?

Pastudent
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    My interpretation would be "functions where M is the domain (and every element of M is mapped)", but the codomain could be anything. – Barry Carter Jul 21 '22 at 13:46
  • and the (codomain is the image space for you?) Meaning, $f : M \rightarrow ?$ Where $?$ means, we don't know where. – Pastudent Jul 21 '22 at 13:48
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    Yes, the codomain is where you have the ?. It's sometimes called the range. The image of f would be a subset of the codomain, potentially the entire codomain – Barry Carter Jul 21 '22 at 13:50
  • The domain is $M$. The codomain (where the values live) is context dependent. – Ethan Bolker Jul 21 '22 at 13:52
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    No, codomain is the set mapped into. Like $f:\mathbb R \to \mathbb R$ defined by $f(x)=x^2.$ This map is into $\mathbb R$ but not onto. Image is non negtive reals. – coffeemath Jul 21 '22 at 13:52
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    Thank you. I usually tend to write everything down formally as much as i can rather than rely on this type of terminology ! – Pastudent Jul 21 '22 at 13:52
  • @barrycarter You mislead the OP when you say codomain sometimes called the range. Only for an onto function is the codomain also the range. – coffeemath Jul 21 '22 at 13:53
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    Depends on the context. In general set theory, this doesn't make much sense, all you know is that the domain is $M$. In real or complex analysis, and to some extent topology, it would likely mean continuous functions $M\to\mathbb R$ or $\mathbb C$. In algebraic geometry, the range could be any field, but the function would have to be regular or rational. – Tom Sharpe Jul 21 '22 at 13:54
  • @TomSharpe That's probably were my question stems from. Since quite often, it is clear from context what is meant. Seldom do the statements come out of the blue and without context, where terminology is the only thing you can rely on. – Pastudent Jul 21 '22 at 13:59
  • Should i close the question, or should i leave it here for other people ? – Pastudent Jul 21 '22 at 14:06
  • @barrycarter, you are free to write it as an anwser, if you like. – Pastudent Jul 21 '22 at 14:07
  • Some people use "function" to mean "function whose codomain is the reals", and reserve "mapping" for functions where the codomain is some other set. I don't like this at all, but I mention it because you may occasionally encounter it. – John Hughes Jul 21 '22 at 14:24
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    @JohnHughes Well, still better than physicists calling distributions functions. – Pastudent Jul 21 '22 at 14:28
  • @coffeemath https://byjus.com/maths/difference-between-codomain-and-range/ notes the 2nd definition of range as It is totally ambiguous and can be used exactly as Codomain showing that range can be used to mean codomain. I was taught to use codomain and image, and to avoid using range because it's ambiguous. – Barry Carter Jul 21 '22 at 14:45
  • I don't believe in answers, but anyone is free to use what I wrote to create an answer. – Barry Carter Jul 21 '22 at 14:45
  • @barrycarter Yes, on further looking I agree with you. It seems "codomain" is unambiguous, as is "image". It was only when teaching intro material from a specific text that I found "range" resticted to mean the image. But I do recall "range" being used in some older texts for the codomain. – coffeemath Jul 21 '22 at 17:05

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In general, it means "functions with domain $M$, but the codomain could be anything."

Often, however, there's context established early in some document, as in, "When we speak of functions in this topology book, we always mean continuous functions." You need to be certain that you're aware of such context. And even the notion of equality of functions can be context-dependent -- sometimes analysts talk about functions being equal when they're equal almost-everywhere.

On a slightly-related note, the sets involved in defining a function have varying names used for them. When you write $f(x)$, the element $x$ is supposed to be in the domain of $f$; the resulting value, $f(x)$ is supposed to be in a set that gets called two different things: some folks call it the "range" (this was popular in the 1960s, when I was learning this stuff), others called it the codomain. If $f$ has domain $D$, we can form the set $S = \{f(x) \mid x \in D \}$. This, too, has two names. Some folks call it the "range", and others call it the image. Because of the name-clash with the other use of "range", it's probably a good idea to stick with "codomain" and "image" and leave "range" out of it.

For a decent description of what a function really is, consider looking at Halmos's Naive Set Theory. When I say "what it really is", I mean "a set theoretic object that has all the properties that we want a function to have, so that we can use it as our notion of a function." There may be other equally good notions of functions that are equivalent to the one in Halmos, of course. Once you have one workable one, you can just go with it, and pretty soon forget the details and just use the properties that you were trying to model.

John Hughes
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