Re: Increasing and decreasing functions - conflicting authors

Dave L. Renfro wrote (in part):

DEFINITION 1: "f is strictly increasing AT x=b" means there
exists delta > 0 such that for all points L
belonging to (b - delta, b) we have f(L) < f(b)
and for all points R belonging to (b, b + delta)
we have f(b) < f(R).

DEFINITION 2: "f is strictly increasing NEAR x=b" means that for
some delta > 0, f is strictly increasing on the
interval (b - delta, b + delta).

DEFINITION 3: "f is strictly increasing on the interval I"
means that for all c and d in I, if c < d,
then f(c) < f(d).

quasi wrote:

Since the phrase "increasing _at_ a point" is already commonly
used to mean what you're calling "increasing _near_ a point",
it would be an uphill fight and cause more confusion to have
a Calculus book use the terminology for "at" and "near" as
you propose above.

Actually, I think you'll find that Definition 1 is much more
common than Definition 2. I know of very few places where
Definition 2 is given, but I could easily (in theory) cite
hundreds (yes, hundreds) of papers and/or texts where
Definition 1 is used. For example, the 1961 paper by
Dvoretzky/Erdös/Kakutani ("Nonincrease everywhere of
the Brownian motion process"), where they prove that
almost every continuous function in the Wiener measure
sense fails Definition 1 at each point.

Probably the analog of Definition 2 is used more in describing
local (modulus-of-continuity) generalized Lipschitz conditions,
although even in this context I think Definition 1 is used
more often. (In the context of Lipschitz conditions,
by the way, Definition 2 is MUCH weaker than Definition 1.)

I would leave "at" to mean definition 2, since that's
pretty standard, and choose some other term for definition 1.

I probably wouldn't want to endorse "near" because it's
too easy to confuse with "at". But in a short discussion
like I gave, it seems useful. Thinking about things some
more, if I were to use "AT" and "NEAR", even in a short
discussion, it'd probably be best to use all capitals
whenever either is being used in the formal sense I defined.

Definition 1 is interesting but rather obscure in the sense
that, for the elementary courses, you don't meet very many
functions which satisfy definition 1 but not definition 2.

I'm not sure what you mean by "elementary", but the example
I gave that distinguishes between them is fairly staid for
an advanced calculus course or an undergraduate real
analysis course. For a typical beginning calculus course,
however, I agree (unless it's an honors class or a class
using Spivak's calculus text or Apostol's calculus text).

The word "at" is the easier and quicker term to conceptualize,
so my choice is to let is be assigned to the concept which
come up more often -- namely definition 2.

I wonder if you got the definitions backwards? Again,
Definition 1 is much more common. It's the notion others
have given in this thread, it's in Spivak's calculus text
(I gave a specific citation in my earlier post), etc.

Dave L. Renfro

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