Re: Increasing and decreasing functions - conflicting authors

In article <TIydnRf1Rq9wYfbZnZ2dnUVZ_sGdnZ2d@xxxxxxxxxxx>,
Colleyville Alan <nospam@xxxxxxxxxx> wrote:
"Arturo Magidin" <magidin@xxxxxxxxxxxxxxxxx> wrote in message
news:e4cosh$g87$1@xxxxxxxxxxxxxxxxxxxxx

[.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.

--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================

Arturo Magidin
magidin@xxxxxxxxxxxxxxxxx

.