Re: Increasing and decreasing functions - conflicting authors

Arturo Magidin wrote (in part):

Second: there is this thing called "convention". For some
authors, for example, a function f is "increasing" on an
interval if and only if for all x,y in the interval, if x<y
then f(x)<f(y); for others, you only require f(x)<=f(y)

Colleyville Alan wrote (in part):

You are bringing up something I had not noticed or thought
of previously.

If the x<y part is left out and only the f(x)<=f(y) is present
in the definition, that seems to imply that the function could
be called "increasing" even if you are reading it "backwards",
i.e. right-to-left. IOW, what is typically shown as a decreaing
interval in many textbooks could be called increasing if you
examined it from right-to-left rather than from left-to-right
e.g. y = -x). Is that the point you were making or did I
totally misinterpret your statement?

Arturo Magidin is talking about the distinction between using
f(x) < f(y), < is "less than", and f(x) <= f(y), <= is "less than
or equal to", in the consequent part of the conditional statements
under discussion. My personal preference is to use "strictly
increasing" for "f(x) < f(y)" and "nondecreasing" (or "monotone
increasing") for f(x) <= f(y)". That is, as much as possible, use
terminology that doesn't force the reader (especially one who
is just looking something up in the book, paper, etc.) to hunt
down the author's usage convention. Incidentally, note that
for "nondecreasing", we can equivalently use "x <= y" for the
antecedent (even in the case of the empty set or a singleton
set for the set the function is to be nondecreasing on).

Dave L. Renfro

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