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...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-02-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2023440?viewType=HTML |
| _version_ | 1797902158320893952 |
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| author | Tianqi Luo Xiaohua Liu |
| author_facet | Tianqi Luo Xiaohua Liu |
| author_sort | Tianqi Luo |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-10T09:13:22Z |
| format | Article |
| id | doaj.art-9cd1659cfe624c2b949b0dde1fba7af6 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-04-10T09:13:22Z |
| publishDate | 2023-02-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-9cd1659cfe624c2b949b0dde1fba7af62023-02-21T01:55:38ZengAIMS PressAIMS Mathematics2473-69882023-02-01848772879210.3934/math.2023440Iteration changes discontinuity into smoothness (Ⅰ): Removable and jumping casesTianqi Luo0Xiaohua Liu1School of Mathematics and Physics, Leshan Normal University, Leshan 614000, ChinaSchool of Mathematics and Physics, Leshan Normal University, Leshan 614000, ChinaIt 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. https://www.aimspress.com/article/doi/10.3934/math.2023440?viewType=HTMLiterationremovable discontinuityjumping discontinuityc<sup>1</sup> smoothpiecewise smooth |
| spellingShingle | Tianqi Luo Xiaohua Liu Iteration changes discontinuity into smoothness (Ⅰ): Removable and jumping cases AIMS Mathematics iteration removable discontinuity jumping discontinuity c<sup>1</sup> smooth piecewise smooth |
| title | Iteration changes discontinuity into smoothness (Ⅰ): Removable and jumping cases |
| title_full | Iteration changes discontinuity into smoothness (Ⅰ): Removable and jumping cases |
| title_fullStr | Iteration changes discontinuity into smoothness (Ⅰ): Removable and jumping cases |
| title_full_unstemmed | Iteration changes discontinuity into smoothness (Ⅰ): Removable and jumping cases |
| title_short | Iteration changes discontinuity into smoothness (Ⅰ): Removable and jumping cases |
| title_sort | iteration changes discontinuity into smoothness i removable and jumping cases |
| topic | iteration removable discontinuity jumping discontinuity c<sup>1</sup> smooth piecewise smooth |
| url | https://www.aimspress.com/article/doi/10.3934/math.2023440?viewType=HTML |
| work_keys_str_mv | AT tianqiluo iterationchangesdiscontinuityintosmoothnessiremovableandjumpingcases AT xiaohualiu iterationchangesdiscontinuityintosmoothnessiremovableandjumpingcases |