Increasing and decreasing functions - conflicting authors
Colleyville Alan
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, or constant intervals as it
is impossible for a point to be increasing and decreasing at the same time.
Tin the Algebra text they show a function that was increasing in the
interval (3,5) and decreasing over the interval (5, inf). Stewart shows a
similar situation. I do not have the book before me, but the gist of it is
a function that is increasing [3,5] and decreasing on [5,inf].
I suspect it is Stewart that is wrong here, but I am not sure. Can anyone
give me a definite answer as to whether such intervals need to be open or
closed?
Thanks.
quasi
This depends on the convention used by a given author. But I'll go
with Stewart's choice since, for example, if you restrict the function
above to the interval [3,5], you get an increasing function.
So we can define the intervals where a function is increasing or
decreasing as the maximal intervals for which the function is
increasing or decreasing.
For example, let f(x)=x^2. Then f is decreasing on (-infinity,0] and f
is increasing on [0,infinity). The fact that 0 is a common point of
both intervals is not a problem precisely because we are talking about
increasing or decreasing on an interval.
quasi
Klueless
> I suspect it is Stewart that is wrong here, but I am not sure. Can anyone give me a definite answer as to whether such intervals
> need to be open or closed?
I think this will be reasonable: (1) You must specify the domain S,
a subset of the reals R, of your real valued function f -- as you should
always do when defining a function. (2) For any point x in S
(A) f is locally "increasing at x" if there is a neighborhood N = (x-e,x+e),
e>0, about x such that for all y in N intersect S
y < x implies f(y) <= f(x)
x < y implies f(x) <= f(y)
(B) f is locally "strictly increasing at x" if there is a neighborhood N = (x-e,x+e),
e>0, about x such that for all y in N intersect S
y < x implies f(y) < f(x)
x < y implies f(x) < f(y)
(3) The function f is globally "increasing" if f is locally "increasing at x" for
all x in S. The function f is globally "strictly increasing" if f is locally
"strictly increasing at x" for all x in S.
(4) The function f is "decreasing at x", "strictly decreasing at x", "decreasing",
"strictly decreasing" if the mirror function g defined by g(z)=f(-z) is
"increasing at -x", "strictly increasing at -x", "increasing", "strictly increasing"
respectively. (Or we might say these terms are just "similarly defined" and
you figure it out.)
Provided I've been careful enough, which I think I have, then parts (3)-(4)
will agree with Definition 4.50 of my Mathematical Analysis, Second Edition
by Apostol, a respected authority.
Note: Here S is any arbitrary subset of the reals, R. S could be open,
closed, or neither in R.
William Elliot
> 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, or constant intervals as it
> is impossible for a point to be increasing and decreasing at the same time.
>
How are they using these terms?
Do they distinguish between increasing and strictly increasing?
> Tin the Algebra text they show a function that was increasing in the
> interval (3,5) and decreasing over the interval (5, inf). Stewart shows a
> similar situation. I do not have the book before me, but the gist of it is
> a function that is increasing [3,5] and decreasing on [5,inf].
>
> I suspect it is Stewart that is wrong here, but I am not sure. Can anyone
> give me a definite answer as to whether such intervals need to be open or
> closed?
It depends upon how the authors are using increasing and decreasing.
Some make the distinction between increasing and strictly increasing as
f is ascending when for all x,y, (x <= y ==> f(x) <= f(y))
f is increasing when for all x,y, (x < y ==> f(x) < f(y))
Similar descending and decreasing, or decreasing and strictly decreasing.
David C. Ullrich
<nos...@nospam.net> wrote:
>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, or constant intervals as it
>is impossible for a point to be increasing and decreasing at the same time.
>
>Tin the Algebra text they show a function that was increasing in the
>interval (3,5) and decreasing over the interval (5, inf). Stewart shows a
>similar situation. I do not have the book before me, but the gist of it is
>a function that is increasing [3,5] and decreasing on [5,inf].
Certainly a function can be increasing on [3,5] and decreasing
on [5, infinity]. The function f(x) = -(x-5)^2 is an example
of such a function.
You seem to be worried about the fact that this seems
to imply that the function is both increasing and
decreasing at the point 5. Don't worry about that.
One speaks of a function being increasing or decreasing
_on_ a _set_.
>I suspect it is Stewart that is wrong here, but I am not sure. Can anyone
>give me a definite answer as to whether such intervals need to be open or
>closed?
>Thanks.
>
************************
David C. Ullrich
Pubkeybreaker
David C. Ullrich wrote:
> On Mon, 15 May 2006 21:35:37 -0500, "Colleyville Alan"
> <nos...@nospam.net> wrote:
>
> You seem to be worried about the fact that this seems
> to imply that the function is both increasing and
> decreasing at the point 5. Don't worry about that.
> One speaks of a function being increasing or decreasing
> _on_ a _set_.
Don't you mean a set with non-zero measure? Or perhaps
a set with more than one element?
The set consisting of the single point 5 is still a set.
Is the function increasing on this set? Or is it decreasing?
Dave Seaman
Yes.
--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
<http://www.mumia2000.org/>
William Elliot
> David C. Ullrich wrote:
> > On Mon, 15 May 2006 21:35:37 -0500, "Colleyville Alan"
> > <nos...@nospam.net> wrote:
>
> > You seem to be worried about the fact that this seems
> > to imply that the function is both increasing and
> > decreasing at the point 5. Don't worry about that.
> > One speaks of a function being increasing or decreasing
> > _on_ a _set_.
>
> Don't you mean a set with non-zero measure? Or perhaps
> a set with more than one element?
>
No, he means (partially) ordered sets. For a function to be increasing,
it's domain and codomain need be ordered sets. For example a function
from a lattice to the reals.
> The set consisting of the single point 5 is still a set.
> Is the function increasing on this set? Or is it decreasing?
The identity function, id:R -> R, x -> x when restriced to any (ordered
inherited) subset of it's ordered domain is still increasing. If
perchance the set A is a singleton or empty, then no matter what the
function f, f restricted to A is increasing and decreasing for A has perse
the trivial order. Thus the technicality of single point domain functions
increasing and decreasing is trivial yet more useful than claiming a line
passing outside thru a corner point is tangent to the corner.
Lee Rudolph
Both, vacuously, given the usual definitions, which begin
(up to irrelevant details) "for all x in X, if y > x then".
Lee Rudolph
Dave L. Renfro
> 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
--------------------------------------------------------
Colleyville Alan
news:Pine.BSI.4.58.06...@vista.hevanet.com...
> On Mon, 15 May 2006, Colleyville Alan wrote:
>
>> 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, or constant intervals as
>> it
>> is impossible for a point to be increasing and decreasing at the same
>> time.
>>
> How are they using these terms?
> Do they distinguish between increasing and strictly increasing?
They do not distinguish. The definition given is:
"A function f is said to be increasing on an open interval l, if for all
a and b in that interval, a < b implies f(a) < f(b)".
The definitions for decreasing and constant intervals are similar.
They go on to explain:
"In calculus, the slope of a line tangent to the graph of a function at
a particular point is used to determine whether the function is increasing,
decreasing, or constant at that point. If the slope is positive, the
function is increasing; if the slope is negative, the function is
decreasing. Since slope cannot be both positive and negative at the same
point, a function cannot be both increasing and decreasing at a specific
point. For this reason, increasing, decreasing, and constant intervals are
expressed in open interval notation."
They go on to mention the example they had just discussed, stating:
"...if [3,5] had been used for the increasing interval and [5,inf) for
the decreasing interval, the function would be both increasing and
decreasing at x=5. This is not possible."
quasi
wrote:
>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, [...]
>
> ...
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.
I would leave "at" to mean definition 2, since that's pretty standard,
and choose some other term for definition 1.
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.
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.
quasi
Colleyville Alan
news:a4dj625frorm9313g...@4ax.com...
> On Mon, 15 May 2006 21:35:37 -0500, "Colleyville Alan"
> <nos...@nospam.net> wrote:
>
>>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, or constant intervals as
>>it
>>is impossible for a point to be increasing and decreasing at the same
>>time.
>>
>>interval (3,5) and decreasing over the interval (5, inf). Stewart shows a
>>similar situation. I do not have the book before me, but the gist of it
>>is
>>a function that is increasing [3,5] and decreasing on [5,inf].
>
> Certainly a function can be increasing on [3,5] and decreasing
> on [5, infinity]. The function f(x) = -(x-5)^2 is an example
> of such a function.
>
> You seem to be worried about the fact that this seems
> to imply that the function is both increasing and
> decreasing at the point 5. Don't worry about that.
> One speaks of a function being increasing or decreasing
> _on_ a _set_.
Well, prior to my College Algebra class last summer, I had no problem with
the idea. It seemed to me that the last element of the set of increasing
points could be the first element in the set of decreasing points. But
Beecher, Peena, & Bittinger were most emphatic. They explained it this way:
"In calculus, the slope of a line tangent to the graph of a function at a
particular point is used to determine whether the function is increasing,
decreasing, or constant at that point. If the slope is positive, the
function is increasing; if the slope is negative, the function is
decreasing. Since slope cannot be both positive and negative at the same
point, a function cannot be both increasing and decreasing at a specific
point. For this reason, increasing, decreasing, and constant intervals are
expressed in open interval notation."
If that is simply a matter of personal preference, I'd like to know. I've
been thinking for the last year that Beecher, et al had corrected my prior
thinking. Now it seems that there is room for other ways of expressing
this.
quasi
<nos...@nospam.net> wrote:
Yes, author's preference. Moreover, most authors do not insist on open
intervals, so it's more like author's prejudice.
I've never heard of the Beecher text, but Stewart is currently pretty
standard nationwide, so perhaps should be given more credence.
quasi
David C. Ullrich
If you want to restrict to sets with more than one element fine.
There's no _need_ to do so, although you get a weird result
if you don't:
Definition: f is strictly increasing on S if f(x) > f(y)
for all x and y in S with x > y.
From that definition it follows that if S has only
one element then yes, f is strictly increasing on
S. Also strictly decreasing.
************************
David C. Ullrich
Dave L. Renfro
>> 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
Arturo Magidin
Colleyville Alan <nos...@nospam.net> wrote:
>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, or constant intervals as it
>is impossible for a point to be increasing and decreasing at the same time.
Okay. First, it is not the ->point<- that is increasing or
decreasing. It is the function which is decreasing.
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), and for the strict inequality you say
"strictly increasing". They are both 'correct' in that there is no
universally agreed upon meaning, and so care must be given to specify
what you mean. There are other definitions.
In fact, under some definitions it would make sense to say a function
is increasing or decreasing at a ->point<-, while in others it would
not make sense.
In short: it is just a matter of preference. As long as the author is
clear and explicit on his or her preference, and the use is consistent
throughout the text, it is fine.
>Tin the Algebra text they show a function that was increasing in the
>interval (3,5) and decreasing over the interval (5, inf). Stewart shows a
>similar situation. I do not have the book before me, but the gist of it is
>a function that is increasing [3,5] and decreasing on [5,inf].
>
>I suspect it is Stewart that is wrong here, but I am not sure.
He is not wrong. He is simply using a different convention.
>Can anyone
>give me a definite answer as to whether such intervals need to be open or
>closed?
There is no definite answer: it will depend on which definitions you
are using. Pick the one that will be used in your course and use that
one.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
mag...@math.berkeley.edu
Dave L. Renfro
> But Beecher, Peena, & Bittinger were most emphatic.
> They explained it this way: "In calculus, the slope
> of a line tangent to the graph of a function at a
> particular point is used to determine whether the
> function is increasing, decreasing, or constant at
> that point. If the slope is positive, the function
> is increasing; if the slope is negative, the function
> is decreasing. Since slope cannot be both positive
> and negative at the same point, a function cannot be
> both increasing and decreasing at a specific point.
> For this reason, increasing, decreasing, and constant
> intervals are expressed in open interval notation."
This is very poor writing (the text, not you).
They make some statements of the form "if the
derivative has a certain sign, then the function
is strictly monotone in a certain way" and then
try to justify something by using the _converses_
of these statements!
Look, here's the simplest way to look at this,
and the way you OUGHT to look at this in a
beginning calculus course:
Increasing and decreasing are concepts that hold
for _intervals_. A function f is (strictly) increasing
on an interval I if c,d in I and c < d implies
f(c) < f(d). This holds for f(x) = x^2 with
I = [0, oo). The derivative is ONE tool that can
help you determine whether or not a function is
increasing on a certain interval. The basic result
is if f' is positive on an open interval, then
f is (strictly) increasing on that open interval.
Whether or not f is strictly increasing on the
corresponding closed interval, or even on a larger
open, closed, or otherwise interval is something
that may or may not happen. For example, f(x) = x^3
is strictly increasing on (-oo, oo), but the only
open intervals that we can automatically conclude
this by using the derivative result are open
subintervals of (-oo, 0) or (0, oo).
In my opinion, textbook authors that insist on
using open intervals run the risk of getting
students confused between the geometric notion
of strictly increasing and the analytic notion
of a a positive derivative.
(For others in this thread.) Let's not forget
strictly increasing singular functions -- functions
that are strictly increasing on an interval and
which have a zero derivative almost everywhere
in that interval in the sense of Lebesgue measure.
Dave L. Renfro
The World Wide Wade
"Colleyville Alan" <nos...@nospam.net> wrote:
> "William Elliot" <ma...@hevanet.remove.com> wrote in message
> news:Pine.BSI.4.58.06...@vista.hevanet.com...
> > On Mon, 15 May 2006, Colleyville Alan wrote:
> >
> >> 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, or constant intervals as
> >> it
> >> is impossible for a point to be increasing and decreasing at the same
> >> time.
> >>
> > How are they using these terms?
> > Do they distinguish between increasing and strictly increasing?
>
>
> They do not distinguish. The definition given is:
> "A function f is said to be increasing on an open interval l, if for all
> a and b in that interval, a < b implies f(a) < f(b)".
> The definitions for decreasing and constant intervals are similar.
>
> They go on to explain:
> "In calculus, the slope of a line tangent to the graph of a function at
> a particular point is used to determine whether the function is increasing,
> decreasing, or constant at that point.
And how do they define "increasing at a point"?
> If the slope is positive, the
> function is increasing;
Actually that's false. You can have f'(a) > 0 without f being
increasing in any open interval containing a, although I expect
these authors couldn't give an example of one, or even understand
the explanation.
> if the slope is negative, the function is
> decreasing. Since slope cannot be both positive and negative at the same
> point, a function cannot be both increasing and decreasing at a specific
> point. For this reason, increasing, decreasing, and constant intervals are
> expressed in open interval notation."
>
> They go on to mention the example they had just discussed, stating:
> "...if [3,5] had been used for the increasing interval and [5,inf) for
> the decreasing interval, the function would be both increasing and
> decreasing at x=5. This is not possible."
An "increasing interval"? An "interval of increase" I can see.
Just another set of daft elementary math text authors.
The World Wide Wade
<1147791695.4...@j73g2000cwa.googlegroups.com>,
"Dave L. Renfro" <renf...@cmich.edu> wrote:
> Increasing and decreasing are concepts that hold
> for _intervals_.
So we can't have increasing functions on the integers?
Dave L. Renfro
>> Increasing and decreasing are concepts that hold
>> for _intervals_.
The World Wide Wade wrote:
> So we can't have increasing functions on the integers?
Sure, and for any subset of the reals. I was talking
about in a beginning calculus course, which I said
just before this in the post you're quoting from.
(For others) We're talking about (well, I am at least)
the global notion of increasing on a set, not about
forming a set by collecting together some or all of
the points of pointwise increase.
Dave L. Renfro
Colleyville Alan
news:waderameyxiii-9B2...@comcast.dca.giganews.com...
They don't define it. This was a College Algebra textbook.
Colleyville Alan
news:4nkj62pje1bfqkuh9...@4ax.com...
Well, Bittinger I know. When I took College Algebra in 1975, the textbook
was by Keedy and Bittinger. So he's been writing textbooks for a long time.
FWIW, this is from his website:
EDUCATION:
BA, Mathematics, Manchester College, 1963
MS, Mathematics, The Ohio State University, 1965
Ph D, Mathematics Education, Purdue University, 1968
EMPLOYMENT:
Professor Emeritus of Mathematics Education,
Indiana University-Purdue University at Indianapolis,
1968 to Present
quasi
<nos...@nospam.net> wrote:
I also know of Bittinger from his basic math text (remedial). Frankly,
in my opinon, Bittinger's book on basic math was horrible.
quasi
Colleyville Alan
news:e4cosh$g87$1...@agate.berkeley.edu...
> In article <jcGdnSJMrYD...@comcast.com>,
> Colleyville Alan <nos...@nospam.net> wrote:
>>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, or constant intervals as
>>it
>>is impossible for a point to be increasing and decreasing at the same
>>time.
>
> Okay. First, it is not the ->point<- that is increasing or
> decreasing. It is the function which is decreasing.
Sloppy wording on my 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)
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?
Dave L. Renfro
>> 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
Arturo Magidin
Colleyville Alan <nos...@nospam.net> wrote:
>"Arturo Magidin" <mag...@math.berkeley.edu> wrote in message
>news:e4cosh$g87$1...@agate.berkeley.edu...
[.snip.]
>> 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)
>
>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?
The latter. To put everything in (wasn't it clear from the syntax of
my sentence?):
Some authors say:
A function f is increasing on an interval if and only if
for all x and y in the interval, if x<y then f(x) < f(y).
Other authors say:
A function f is increasing on an interval if and only if
for all x and y in the interval, if x<y then f(x) <= f(y).
Which was, I think, extremely clear from the rest of the paragraph
after you clipped, where I talked about "strictly increasing" and so
on.
Colleyville Alan
news:1147955490.6...@j73g2000cwa.googlegroups.com...
> 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.
Thanks, I missed that pesky little equals sign!
I imagine Andy Rooney would say "did you notice that it's hard to do
mathematics if you're dyslexic?"
Colleyville Alan
It may have been extremely clear, but it seems I never noticed the equals
sign. Pattern recognition is not my strong suit and I have a tendency
towards dyslexia as well (actually "expressive disgraphia", but close
enough). Frankly, I'm glad that my interpretation was wrong. It'd be a
little too counter-intuitive to try and learn math otherwise.
Also, thanks to everyone who replied. I was not sure what the "strictly"
refered to and now it is clear.