@article{Luo2023, 
author = {Tianqi Luo and Xiaohua Liu},
title = {Iteration changes discontinuity into smoothness (Ⅱ): oscillating case},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {4},
pages = {8793-8810},
keywords = {iteration, oscillating discontinuity, C1 smooth, piecewise C1 smooth},
url = {https://www.sciopen.com/article/10.3934/math.2023441},
doi = {10.3934/math.2023441},
abstract = {It has been shown that a self-mapping with exactly one removable or jumping discontinuity may have a        C    1   smooth iterate of the second-order. However, some examples show that a self-mapping with exactly one oscillating discontinuity may also have a        C    1   smooth iterate of the second-order, indicating that iteration can turn a self-mapping with exactly one oscillating discontinuity into a        C    1   smooth one. In this paper, we study piecewise        C    1   self-mappings on the open interval    (  0  ,  1  ) having only one oscillating discontinuity. We give necessary and sufficient conditions for those self-mappings whose second-order iterates are        C    1   smooth.}
}