From: lerma@math.nwu.edu (Miguel A. Lerma)
Subject: Re: continuity of the derivative
Date: 20 Jul 2000 03:44:02 GMT
Newsgroups: sci.math
Summary: [missing]

artur.steiner@usa.net wrote:
: Do you agree that if a real valued function f is differentiable on the
: interval [a , b] and its derivative f' is monotone there then f' is
: also continuous on [a , b]? This is true because the derivative f'
: satisfies the Intermediate Value Property, right?

The result follows from the fact that the derivative cannot
have jump discontinuities at points where it is defined. 
In fact, assume that f' has a jump of length L != 0 at x. 
Let h be some small positive number. Then, by the mean value 
theorem:

     ((f(x+h) - f(x))/h = f'(x + u h)

     ((f(x) - f(x-h))/h = f'(x - v h)

where u, v are between 0 and 1. If f'(x) exists then the left hand 
sides tend to f'(x) as h -> 0, and their difference can be made as 
small as we want by taking h small enough. However the difference
between the right hand sides tend to L, a contradiction.

Since a monotone function can have jump discontinuities only,
it follows that if f' is monotone then it must be continuous.


Miguel A. Lerma