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Discontinuities and smooth curves in n-space
by
Peter Nyikos
University of South Carolina
The calculus textbook used by our department has the following theorem, which clearly extends to all n ≥ 2:
Theorem 1. If the limit of a real-valued function on R2 exists at a point p, then it will also be the limit along any smooth curve through p.
Depending on whether or not "smooth" includes the requirement that the curve be regular, i.e., have a parametrization for which the derivative is never the zero vector, we have either Theorem 2 or Theorem 3 below.
Theorem 2. If f is a real-valued function defined in a deleted neighborhood of p in Rn, and the limit of f at p does not exist, then there is a (not necessarily regular) smooth curve through p on which the limit does not exist.
Theorem 3. If f is a real-valued function defined in a deleted neighborhood of p in Rn, and the limit of f at p does not exist, then either (1) there is a regular smooth curve through p on which the limit does not exist, or (2) there are two straight lines through p on which the limits exist, but are unequal. Moreover, if (1) fails, and r and r' are distinct limits along two straight lines, then every real number between r and r' is the limit along some straight line through p.
(1) can indeed fail: consider the unit ball in R3, endowed with parallels of latitude like our earth, 0 at the equator and 90 at the poles (but no distinction between north and south, so that antipodes get the same value). Extend this real-valued function from the unit sphere inward, except to the very center point p, so that the resulting f is constant on each diameter (except for not being defined at the center p).
It is easy to see that on any regular smooth curve through p, this function approaches the value along the tangent line to the curve.
The proof of Theorem 3 (of which Theorem 2 is an easy corollary) has a nice classical set-theoretic-topology component.
Date received: February 28, 2008
Copyright © 2008 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cawy-25.