Re: Increasing and decreasing functions - conflicting authors

Colleyville Alan wrote (in part):

I am starting to read Stewart's Calculus Early Transcendentals
(1999 ed.) for a Calc I class that will begin in a few weeks.
In it he talks about increasing and decreasing functions and
shows the intervals using closed interval notation. In my
College Algebra text (Beecher, Penna, & Bittinger), they
emphatically state that you need to use open interval notation
when discussing increasing, decreasing, [...]

This comes up at least every couple of months in the
AP-Calculus list group (archived at The Math Forum),
and I'd wager that it generates nearly as much discussion
there as 1 = 0.999... posts generate in sci.math.

Below is my most recent post on this topic, in case anyone here
is interested. That post is not intended to be at the level of
the original poster's question in the present sci.math thread,
by the way. Following this post on pointwise notions of increase
and decrease is a typical example of the confusion I sometimes
see between "derivative is positive on an interval" and
"increasing on an interval".

--------------------------------------------------------

http://mathforum.org/kb/message.jspa?messageID=4661014

Lin McMullin wrote (in part):

http://mathforum.org/kb/message.jspa?messageID=4658938

Any "definition" of increasing at a point would
have to include something implying an interval
(e.g. "as the function moves through the point"
or "in some open interval around the point,"
or some such). I know of no book that defines
this phrase.

Spivak's beginning calculus book (below) has a long problem,
#65 (a) through (f) in Chapter 11 (pp. 214-215), that deals
with the idea "f is increasing at a" in the same sense that
I use below (Definition #1).

Michael Spivak, "Calculus", 3'rd edition, Publish or Perish,
1994, xiv + 670 pages. ISBN 0-914098-89-6

This idea occurs in many undergraduate and graduate level
real analysis texts and it's a well-known and useful idea
in mathematical research. More generally, this idea is one
instance of a general notion that is sometimes called
"localization at point". For functions, you can form the
pointwise version of any interval property (increasing,
concave up, etc.) by requiring the property to hold on all
sufficiently small intervals centered at the point (this is
Definition #2 below). The idea of monotonicity at a point is
also used extensively in certain areas of probability and
statistics, such as in the analysis of Brownian motion.

Without getting into other variations that one also encounters
(such as local monotonicity variations that arise by only
requiring that the restriction of the function is monotone
relative to (all sufficiently small) types of sets which
are not necessarily full intervals), here are the notions
of monotonicity that I've found to be the most commonly used,
specialized to the case of "strict increase".

In increasing order of strength, they are:

* strictly increasing at a point

* strictly increasing near a point

* strictly increasing on a specified interval

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).

Sometimes the phrase "locally increasing" is used for Definition #2,
but since this phrase is also often used for Definition #1, I'll
use the word "near" in order to distinguish them. (I have not
seen "at" and "near" used to distinguish these two concepts
before, but this seems to be a nice way to distinguish them.)

THEOREM 4: Let I be an open interval. The following are equivalent:

(1) f is strictly increasing at each point of I.
(2) f is strictly increasing near each point of I.
(3) f is strictly increasing on I.

Proof: (3) ==> (2) ==> (1) is immediate, and the proof
of (1) ==> (3) involves a compactness argument
(not compactness of I, but compactness of the closed
interval [c,d], where c < d are the two points
arbitrarily chosen in I during the process of proving
that f is strictly increasing on I).

Theorem 5: Let f be a function defined on an open interval I
containing b. Then for x=b we have (3) ==> (2) ==> (1),
but neither of these two implications is reversible.

Proof: Again, (3) ==> (2) ==> (1) is immediate. (2) doesn't
imply (3): sin(x) is increasing near x=0 but sin(x)
is not increasing on the interval (-10, 10).
(1) doesn't imply (2): Define the function f by
f(x) = x + (x^2)*sin(1/x^2), with f(0) = 0.

Then f is strictly increasing at x=0 (in fact, f' exists and
equals 1 at x=0), but f has infinitely many intervals of strict
increase and strict decrease arbitrarily close to x=0 (on both
sides of x=0 in fact), and hence f isn't strictly increasing
near x=0 (or even non-decreasing near either side of x=0).

The relationship between these notions and pointwise
differentiation notions (specifically, the four Dini
derivates that one encounters in beginning graduate level
real analysis classes) is a little involved, but one general
theme in this relationship is that, roughly, the sign of
the differentiation notion corresponds to a pointwise
strict monotonicity notion (Definition #1 above) along
with a lower bound on how rapidly the function increases
or decreases at that point. For example, the function x^3
is increasing at x=0, but not with sufficient rapidity
at x=0 for its derivative to be positive at x=0.

Dave L. Renfro

--------------------------------------------------------

Jim Rahn wrote:

http://mathforum.org/kb/message.jspa?messageID=4671741

The endpoints are not usually graded. We have
usually allowed the students to say the function
is increasing on the interval (2,4) OR [2,4].
I personally allow my students to write this
because it doesn't make sense. If f'(2)=f'(4)=0
and f'(c)>0 for all c in (2,4) then I would
write f is increasing on the interval (2,4).

I don't see why this doesn't make sense, unless
you're reading the question as "on what intervals
is the derivative positive", instead of "on what
intervals is the function increasing". Also, from
my grading experience, I was under the impression
that the endpoints are _never_ graded (not just
"not usually"), at least not when the function
is defined and appropriately one-sided continuous
at the endpoints.

Dave L. Renfro

--------------------------------------------------------

.

Loading