From: ags@seaman.cc.purdue.edu (Dave Seaman) Newsgroups: misc.education,sci.math Subject: Re: Only 40% of teachers KNOW what they teach?? Date: 8 Sep 1998 12:29:17 -0500 In article , David Kastrup wrote: >Darrell Ryan writes: > >> differentiability implies continuity. > >But not of the derivative. For example, > >x^2 cos(1/x) for x /= 0, 0 for x=0 is differentiable everywhere, but the >derivative is not continuous at x=0. Yes, but that function still has the crucial property needed to make the MVT (actually Rolle's Theorem) work: it is differentiable everywhere, and therefore the derivative is zero at each extreme point (x=0 is not an extreme point for that function -- not that it matters, since f'(0) = 0 anyway). MVT requires only that f be continuous on [a,b] and differentiable on (a,b). It doesn't require the derivative to be continuous. -- Dave Seaman dseaman@purdue.edu ++++ stop the execution of Mumia Abu-Jamal ++++ ++++ if you agree copy these lines to your sig ++++ ++++ see http://www.xs4all.nl/~tank/spg-l/sigaction.htm ++++