Re: Increasing and decreasing functions - conflicting authors

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?

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.

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