Re: Increasing and decreasing functions - conflicting authors
- From: magidin@xxxxxxxxxxxxxxxxx (Arturo Magidin)
- Date: Tue, 16 May 2006 14:50:57 +0000 (UTC)
In article <jcGdnSJMrYDhp_TZRVn-iA@xxxxxxxxxxx>,
Colleyville Alan <nospam@xxxxxxxxxx> 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
magidin@xxxxxxxxxxxxxxxxx
.
- Follow-Ups:
- Re: Increasing and decreasing functions - conflicting authors
- From: Colleyville Alan
- Re: Increasing and decreasing functions - conflicting authors
- References:
- Increasing and decreasing functions - conflicting authors
- From: Colleyville Alan
- Increasing and decreasing functions - conflicting authors
- Prev by Date: Re: brownian motion
- Next by Date: Is there an example of a function which ...
- Previous by thread: Re: Increasing and decreasing functions - conflicting authors
- Next by thread: Re: Increasing and decreasing functions - conflicting authors
- Index(es):
Relevant Pages
|
