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Simultaneous extension of continuous and uniformly continuous functions

Volume 265 / 2022

Valentin Gutev Studia Mathematica 265 (2022), 121-139 MSC: 54C20, 54C30, 54C35, 54E40, 46B40, 26A16. DOI: 10.4064/sm201019-22-6 Published online: 24 February 2022


The first known continuous extension result was obtained by Lebesgue in 1907. In 1915, Tietze published his famous extension theorem generalising Lebesgue’s result from the plane to general metric spaces. He constructed the extension by an explicit formula involving the distance function on the metric space. Thereafter, several authors produced other explicit extension formulas. In the present paper, we show that all these extension constructions also preserve uniform continuity, which answers a question posed by St. Watson. In fact, such constructions are simultaneous for special bounded functions. Based on this, we also refine a result of Dugundji by constructing various continuous (nonlinear) extension operators which preserve uniform continuity as well.


  • Valentin GutevDepartment of Mathematics
    Faculty of Science
    University of Malta
    Msida MSD 2080, Malta

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