DATE(mo/da/yr):   10/31/1997

SPEAKER_OR_HOST: Andrew M. Bruckner

PERIOD_OF_VISIT: 

LOCATION:  K9509

TIME(hr:min[a|p]): 11:30--12:30

TITLE:  The X, Y and Z's of Characterizing Derivatives

AFFILIATION: University of California, Santa Barbara

ABSTRACT:  

Most important classes of functions admit a variety of characterizations
- that is, theorems that could provide alternate definitions for the
class. For example, complex analytic functions defined on a disk are
usually defined as the differentiable ones, but could equally well be
defined as those possessing a Taylor expansion, or via the
Cauchy-Riemann equations, or by integrals being independent of the path
or via conformal maps, etc.

Simarly, there are many useful characterizations for classes of real
functions (e.g. the continuous functions, measurable functions,
absolutely continuous functions, etc.). Obtaining useful
characterizations for the class of derivatives - that is, functions that
ARE (not HAVE) derivatives - has been difficult.

We review the history of this characterization problem, indicate the
kinds of characterizations that are sought, discuss some of the failed
attempts, criticize some of the 'successful' attempts, and indicate how
even the failed attempts have added greatly to our understanding of
derivatives. We also try to provide insight to the question, "Why is this
problem so difficult?"

We aim the talk to a 'general' audience. Most technical details will be
relegated to peripheral comments.