Discontinuous, although “highly” differentiable, real functions and algebraic genericity

We exhibit a class of functions f:R→R which are bounded, continuous on R∖Q, left discontinuous on Q, right differentiable on Q, and upper left Dini differentiable on R∖Q. Other properties of these functions, such as jump sizes and local extrema, are also discussed. These functions are constructed us...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2021-10, Vol.502 (2), p.125264, Article 125264
Main Authors: Adell, J.A., Falcó, J., Rodríguez-Vidanes, D.L., Seoane-Sepúlveda, J.B.
Format: Article
Language:English
Subjects:
Differentiable function
Lineability
Local extrema
Poisson process
Probabilistic function
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Summary:We exhibit a class of functions f:R→R which are bounded, continuous on R∖Q, left discontinuous on Q, right differentiable on Q, and upper left Dini differentiable on R∖Q. Other properties of these functions, such as jump sizes and local extrema, are also discussed. These functions are constructed using probabilistic methods. We also show that the families of functions satisfying similar properties contain large algebraic structures (obtaining lineability, algebrability and coneability).
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2021.125264