Re: IN, ON, OVER

In article <1147913564.499527.184630@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
R. Colacitti <wwromeo@xxxxxxxxx> wrote:

Just looking for some clarification on the uses of "in" "on" and "over"
when talking about properties of functions or variables in theorems.

For ex ... I see this in a book ...

"If a function f is continuous on [a, b] and f'(x) = 0 for all x in
(a,b) ... then..."

When can one use "on" and when can't they use it?

I don't see why that statements doesn't say f'(x) or f' = 0 on (a,b).

It is quite correct, and often done, to say "f is continuous in [a,b]"
or "f' = 0 on (a,b)".
On the other hand, you shouldn't say "f'(x) = 0 on (a,b)", because
the phrase "on (a,b)" doesn't mention the variable x. You also
shouldn't say "for all x on (a,b)": variables are "in", not "on",
sets, while functions can have some property either "on" a set or
"in" a set...

I also see some theorems that talk about properties "over" some set or
interval.

.... and sometimes "over" a set, though this is a bit less common
I think. A specific usage of "over" in algebra is to designate
the field or ring of scalars for a vector space or module.

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

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