0
$\begingroup$

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?

  • 0
    Oh. I don't think that happens. The converse does, but maybe not in spaces that calculus is concerned with.2012-05-22

1 Answers 1

2

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.

  • 0
    I have edited my description.2014-11-13