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A New, Harder Proof That Continuous Functions With Schwarz Derivative 0 Are Lines
 

Summary: A New, Harder Proof That Continuous Functions With
Schwarz Derivative 0 Are Lines
J. Marshall Ash
Abstract. The Schwarz derivative of a real-valued function of a real variable
F is de...ned at the point x by
lim
h!0
F (x + h) 2F (x) + F (x h)
h2
:
The usual proof that a function with identically 0 Schwarz derivative must be a
line depends on the fact that a continuous function on a closed interval attains
a maximum. Here we give an alternate proof which avoids this dependence.
1. Introduction
The Schwarz derivative is de...ned to be
(1) DF (x) := lim
h!0
F (x h) 2F (x) + F (x + h)
h2
:

  

Source: Ash, J. Marshall - Department of Mathematical Sciences, DePaul University

 

Collections: Mathematics