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Topic: singular differentiable strictly increasing function
Replies: 27   Last Post: May 7, 2012 5:22 PM

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 bacle Posts: 839 From: nyc Registered: 6/6/10
Re: singular differentiable strictly increasing function
Posted: May 3, 2012 2:38 AM

> On May 3, 4:57 am, Gus Gassmann
> wrote:

> > On May 2, 10:37 pm, Tonico <Tonic...@yahoo.com>
> wrote:
> >
> >
> >
> >
> >

> > > On May 2, 9:30 pm, boyandshark
> <amitgan...@gmail.com> wrote:
> >
> > > > Looking around at various sources, I could not
> quite find an answer to
> > > > the following issue:.
> >
> > > >  Suppose we have a differentiable real value
> function that is strictly
> > > > increasing over its domain. Do these conditions
> this imply that the
> > > > derivative is almost everywhere non-zero? I
> know there exist counter-
> > > > examples to this claim if the function is only
> almost everywhere
> > > > differentiable - hence I was wondering if it
> was possible that the
> > > > strengthening to everywhere differentiable
> might remove these
> > > > pathologies?
> >
> > > > Thanks immensely for any feedback.
> >
> > > Not only different from zero: if a differentiable
> (on its domain)
> > > function (one real variable and real valued one)
> is strictly
> > > increasing in its domain then its derivative is
> actually positive.
> >
> > This is certainly not true: What about f(x) = x^3?-

>
>
> Indeed, inflexion points are problematic, and
> checking the example one
> could ask whether there can be "lots of these points"
> as to make the
> derivative to be zero in aset with non-zero measure.

A monotone function is a.e. differentiable. A monotonous function, who knows...

> My bet is on
> "NO", but taking into account some weird things in
> analysis perhaps
> someone can come up with a counterexample...
>
> Tonio