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

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 Butch Malahide Posts: 885 Registered: 6/29/05
Re: singular differentiable strictly increasing function
Posted: May 4, 2012 1:55 PM

On May 4, 11:16 am, Tonico <Tonic...@yahoo.com> wrote:
> On May 4, 5:42 pm, David C. Ullrich <ullr...@math.okstate.edu> wrote:
>

> > Inflection points are far from the only way a strictly increasing
> > differentiable function can have vanishing derivative at a point.

>
> I bet they are, but that's what popped up when someone showed me wrong
> in my first answer to the OP.
>
> Perhaps you can add some other examples...?

What about the example I posted early in this thread, of a strictly
increasing differentiable function f such that the set {x: f'(x) > 0}
has measure less than one? (Obviously, you could make the measure of
that set arbitrarily small, but that's beside the point.) It follows
that the set of points where f' vanishes has infinite measure, and is
therefore uncountable. Since the set of inflexion points is at most
countable, you have uncountably many points where f' vanishes but
which are not inflexion points. (You could carry out the construction
so that there are no inflexion points at all.)

> Still, the measure of the set of points where such an strictly
> increasing function has a vanishing (non-vanishing) derivative remains
> open, me believes.