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This is a trivial vocabulary question.

It seems to me that "constant on every connected component of the domain" would be a reasonable definition of the term "componentwise constant", provided that there's not already some other concept conventionally denoted by that same expression.

The term, in quotes, gets a lot of google hits, suggesting it does conventionally refer to something. Is that it, or is it something else?

If something else, then what term should be used instead? If $f'=0$, then $f$ is $\cdots\cdots\cdots\cdots\cdots\cdots\cdots$. If $f'=g'$, then $f$ and $g$ differ by a $\cdots\cdots\cdots\cdots\cdots\cdots\cdots$ function.

Later note: Someone in the "comments" section below proposed "locally constant". I think that's mistaken, for reasons I explained there. The comment got five up-votes. Are those people confused or am I?

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    I think what you are calling "componentwise constant" is usually called "locally constant."2012-05-06
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    In my experience *componentwise* generally refers to the components of points in a Cartesian product; e.g., ordinary vector addition is defined componentwise.2012-05-06
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    @froggie : I think you're mistaken. The identity function on the rational numbers is constant on connected components of its domain (each connected component contains just one point) but is it "locally constant"?2012-05-07
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    @Michael Hardy: you're absolutely right, locally constant is different than what you are saying. Silly mistake, but an easy one to make, because in many situations that come up in practice, the notions do coincide. Thanks for pointing this out!2012-05-07
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    The other thing that bugs me about bringing in the word "locally" is that this is a sort of "global" thing. "Locally" should mean something like "for every open neighborhood of every point....". The proof of the propositions in calculus mentioned in the question involve the mean value theorem, which goes from a _local_ hypothesis to a _global_ conclusion (slopes of secant lines are "global").2012-05-07
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    Whie locally constant is technically incorrect in general, I have to wonder what the situation is here that would make a separate term necessary...2012-05-07
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    @HarryAltman : What would make a separate term appropriate is that "locally constant" could mislead students and others.2012-05-08
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    I meant, I was wondering what the situation was that was causing such functions (that are constant on connected components, but not necessarily locally constant) to come up in a space which isn't locally connected...2012-05-08
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    @HarryAltman : Probably only locally connected spaces would be involved, but even so, that terminology could confuse and mislead.2012-05-08
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    I think the point is that **locally constant** is a natural and useful notion (e.g. it correctly fills in the blanks in your question above). You're quite right that it does not agree with "constant on connected components" on a general space...but what is a natural situation in which we are interested in functions which are constant on connected components but not locally constant??2012-05-22
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    @PeteL.Clark : I still think the term could mislead the reader as to what the word "locally" should normally mean, and the reader who already knows what it means might get the wrong idea about what is intended.2012-05-22
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    @Michael: I am not suggesting to use the phrase "locally constant" for "constant on connected components". I am asking for an example in which "constant on connected components but not locally constant" naturally arises.2012-05-22
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    Oh. I don't think that happens. The converse does, but maybe not in spaces that calculus is concerned with.2012-05-22

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Here is my take on the question and discussion:

'Component-wise constant' is not a widely used term. Googling the term shows that it is in use, but possibly by a fairly small group of writers. Without searching through papers to find the definition, it appears that the usage is 'constant on the components of the complement of some measure-zero set', or something like that.

In fact, there is no widely-recognised term that matches the required definition. It is unlikely that creating a new term would be a good plan in this case. On that basis, the term 'locally constant' has been considered as an alternative. This term does not have exactly the meaning required, but is generally understood and gives the same outcome in the cases that are likely to arise.

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    Why is "constant on connected components" not an option?2014-11-13
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    I'd say it is an option, but the question sounds like the OP wants something shorter.2014-11-13
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    They are components of the _domain_, not of the graph, thus not of a Cartesian product. ${}\qquad{}$2014-11-13
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    @MichaelHardy Yes, I think you're right. I didn't do enough digging. I was misled by the paper I checked being about componentwise-constant things on a Cartesian product.2014-11-13
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    I have edited my description.2014-11-13