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If a field L extends a subfield K then L is called an extension of K and K is called the extension's base field. See extension field for a definition.

What is the analog of "base field" when it's a function being extended?

That is, if a function L extends a given function K then L is called an extension of K and the original function K is called the extension's ____?.

The word "restriction" does not work because no one calls K "the extension's restriction" and I don't want to make up new terminology that no one else uses. Whenever the word "restriction" is applied to a function, then the set that it is being restricted to is always either explicitly mentioned or else clear from context. For example, people often write "the function's restriction to this-or-that set", and if the set is clear from context then they might simply refer to it as "the function's restriction". But (at least based on my Internet searches), no one writes "the function's restriction" without somehow making clear the set that it is being restricted to.

Example usage:

If an extension has the same [this-or-that property] as its ______ then they will also have the same [this-or-that property].

What word fits in the blank?

More generally, we have a noun ("extension") for any thing produced by an act of extending. Is there a noun for the thing that is going to be extended (before the act occurs)?

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    $\begingroup$ What's the matter with base function? If you substitute function for field in the definition, why not in the name, too? $\endgroup$
    – John Lawler
    Feb 15, 2023 at 23:06
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    $\begingroup$ "I don't want to make up new terminology that no one else uses." You'll likely find a better answer on math.SE. This isn't about English per se, but about the jargon used in a particular field. $\endgroup$
    – jimm101
    Feb 15, 2023 at 23:07
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    $\begingroup$ I’m voting to close this question because it belongs in math.stackexchange.com $\endgroup$
    – Greybeard
    Feb 15, 2023 at 23:18
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    $\begingroup$ Although I have large exposure to people writing about function extensions, I haven't seen a specific term for that. Either people use everyday English (the original function, for example), or use the name of the function (usually, the extension of $f$ is denoted by $\overline{f}$, $\tilde{f}$ or something similar, to make things simpler). $\endgroup$ Feb 16, 2023 at 0:28
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    $\begingroup$ If there is a standard term, then math.stackexchange.com would be better. But perhaps there is no standard term, and it sounds like you're trying to help your students understand by writing well about this. In that case, I think it does belong here. $\endgroup$
    – Sue VanHattum
    Feb 16, 2023 at 3:19

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I'd just say "the original function", having a pretty good colloquial sense, rather than create a technical word. Yes, if you are considering such stuff in a more extravagant way (e.g., in a vast category-theoretic setting), you might have a need to formalize this relationship, but, in my own experience, just the colloquial language is adequate. Or, in some modest examples, "restriction of the extension", but I've used this in situations where the "restriction" was not the original thing... For example, fooling around with unbounded self-adjoint operators (already in Sturm-Liouville problems, so it's not at all contrived).

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This might answer your question in a backwards way.

We have the notion of germ equivalence of two functions. Given a point $x$, we say that functions $f$ and $g$ define the same germ at $x$ if there is a neighborhood $U$ of $x$ (open set containing $x$) with $f\vert_{U}=g\vert_U$

Germ equivalence focuses on localization, as opposed to extension which is more of a global focus.

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