From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik)
Subject: Re: Please help me show that...
Date: 30 Mar 1999 09:55:26 -0500
Newsgroups: sci.math
Keywords: derivatives have the Darboux property

In article <3700428B.80361888@pilot.msu.edu>,
Roy Klescw..... <klesczew@pilot.msu.edu> wrote:
:Show that if  f(x) is a function whose derivative f `(x) is a monotonic,
:then f `(x) is continuous.
:Thanks in advance
:Roy
:
 (1) If a function has a derivative everywhere in an interval then the
derivative, whether continuous or not, has the Intermediate Value Property
(also called Darboux Property): the image of every interval is an
interval. The proof is rather standard, like the proof of Rolle's Theorem.
 (2) A monotonic function can have only one type of discontinuity: a jump.

 Punchline: Can a monotonic function with Darboux Property have a jump?

Hope it helps, ZVK(Slavek).