Open Discontinuous Maps From ℝⁿ onto ℝⁿ

Abstract

In 1962 Spira found an example of open map from ℝⁿ onto ℝⁿ that is discontinuous at a Cantor set. Subsequent published examples of everywhere discontinuous open maps from ℝⁿ onto ℝⁿ are either nonmeasurable, not computable, or difficult to visualize. We provide a simple example. Published examples of open discontinuous maps from ℝⁿ onto ℝⁿ are discontinuous at infinitely many points and are infinite-to-one. We give two more examples of open discontinuous maps from ℝⁿ onto ℝⁿ; the first is discontinuous at only one point, and the second is at most two-to-one.