@article{Luo2023, 
author = {Tianqi Luo and Xiaohua Liu},
title = {Iteration changes discontinuity into smoothness (Ⅰ): Removable and jumping cases},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {4},
pages = {8772-8792},
keywords = {iteration, removable discontinuity, jumping discontinuity, C1 smooth, piecewise smooth},
url = {https://www.sciopen.com/article/10.3934/math.2023440},
doi = {10.3934/math.2023440},
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.}
}