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

Tianqi Luo; Xiaohua Liu; Tianqi Luo; Xiaohua Liu

doi:10.3934/math.2023440

AIMS Mathematics

2023, Volume 8, Issue 4: 8772-8792. doi: 10.3934/math.2023440

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Research article

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

Tianqi Luo ,
Xiaohua Liu ^,

School of Mathematics and Physics, Leshan Normal University, Leshan 614000, China

Received: 24 March 2022 Revised: 27 May 2022 Accepted: 31 May 2022 Published: 08 February 2023
MSC : 37E05, 39B12

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.
- iteration,
- removable discontinuity,
- jumping discontinuity,
- $ C^{1} $ smooth,
- piecewise smooth
Citation: Tianqi Luo, Xiaohua Liu. Iteration changes discontinuity into smoothness (Ⅰ): Removable and jumping cases[J]. AIMS Mathematics, 2023, 8(4): 8772-8792. doi: 10.3934/math.2023440

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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.

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