derivative of strictly increasing continuous function

[Li Yi]

Let f be a strictly increasing continous function. Is it true that if f
is not differentiable at x0, it must hold that f'(x0) = infinity? (that
is, is it true that lim (f(x+h)-f(x))/h exists or be infinity?)

[Robert Israel]

In article <1165207331.1...@80g2000cwy.googlegroups.com>,

No. Given any sequence m_n in (0,infty), it's easy to
construct an example where there is a sequence x_n -> x_0 with
(f(x_n) - f(x_0))/(x_n - x_0) = m_n.

Robert Israel isr...@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

[drmw...@gmail.com]

How about f(x) given by x for x<0, and 2x for x>=0?

It's continuous everywhere, but non-differentiable at x=0 (and not in
the sense of infinity).

Dr. Michael W. Ecker
Associate Professor of Mathematics
Pennsylvania State University
Wilkes-Barre Campus
Lehman, PA 18627

[Dave L. Renfro]

Li Yi wrote:

Let d- & d+ be the lower left & lower right Dini derivates
and D- & D+ be the upper left & upper right Dini derivates
(of some specified function).

You appear to be asking if either d- = d+ = D- = D+ = finite
or d- = d+ = D- = D+ = infinity must hold at each point of
a strictly increasing continuous function.

In fact, given any two extended real numbers r and s such that
0 <= r < s <= oo, there exists a strictly increasing continuous
function such that we have d- = d+ = r < s = D- = D+ for a
co-meager (but necessarily of measure zero, of course) set of
real numbers. In particular, this can hold at c-many points in
every open interval (c = cardinality of the reals). I'm also
pretty sure (but not certain; I'd have to look over some
constructions/proofs that I don't have with me right now) that,
for a strictly increasing continuous function, we can have
0 <= d- < d+ = r < s = D- < D+ <= oo on a smaller type of set
(a certain subclass of the sigma-porous sets) that allows for
examples having Hausdorff dimension 1 in every open interval
(in particular, it can hold at c-many points in every open
interval).

ESSAY ON NON-DIFFERENTIABILITY POINTS OF MONOTONE FUNCTIONS
http://groups.google.com/group/sci.math/msg/1bd39d992c91e950

Dave L. Renfro