Iteration changes discontinuity into smoothness (Ⅰ): Removable and jumping cases

Abstract

It has been proved that a self-mapping with exact one discontinuity may have a continuous iterate of the second order. It actually shows that iteration can change discontinuity into continuity. Further, we can also find some examples with exact one discontinuity which have $ C^1 $ smooth iterate of the second order, indicating that iteration can change discontinuity into smoothness. In this paper we investigate piecewise $ C^1 $ self-mappings on the open interval $ (0, 1) $ having only one removable or jumping discontinuity. We give necessary and sufficient conditions for those self-mappings to have a $ C^1 $ smooth iterate of the second order.

Keywords

• iteration
• removable discontinuity
• jumping discontinuity
• c¹ smooth
• piecewise smooth