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

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.