# “componentwise constant”?

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." – froggie May 6 '12 at 17:41
In my experience componentwise generally refers to the components of points in a Cartesian product; e.g., ordinary vector addition is defined componentwise. – Brian M. Scott May 6 '12 at 17:42
@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"? – Michael Hardy May 7 '12 at 18:13
@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! – froggie May 7 '12 at 18:28
Whie locally constant is technically incorrect in general, I have to wonder what the situation is here that would make a separate term necessary... – Harry Altman May 7 '12 at 20:02

## 1 Answer

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? – Najib Idrissi Nov 13 '14 at 20:07
I'd say it is an option, but the question sounds like the OP wants something shorter. – Jessica B Nov 13 '14 at 20:11
They are components of the domain, not of the graph, thus not of a Cartesian product. ${}\qquad{}$ – Michael Hardy Nov 13 '14 at 21:10
@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. – Jessica B Nov 13 '14 at 21:23
I have edited my description. – Jessica B Nov 13 '14 at 21:35