Every set is ``symmetric'' for some function

Year, pages: 2013, 897 - 901

Let $\langle h_n\rangle$ denote a sequence of positive real numbers. We show that for every set $A \subseteq \mathbb{R}$ there exists a function $f:\mathbb{R}\toω$ such that $A = \{x\in\mathbb{R}: (\exists \langle h_n\rangle) [h_n \searrow 0 & (\forall n\in\mathbb{N})(f(x - h_n)= f(x + h_n) = f(x))]\}$. This solves a problem of K. Ciesielski, K. Muthuvel and A. Nowik.

Nowik, A., Szyszkowski, M. 2013. Every set is ``symmetric'' for some function. In Mathematica Slovaca, vol. 63, no.4, pp. 897-901. 0139-9918. DOI: https://doi.org/10.2478/s12175-013-0143-0

Nowik, A., Szyszkowski, M. (2013). Every set is ``symmetric'' for some function. Mathematica Slovaca, 63(4), 897-901. 0139-9918. DOI: https://doi.org/10.2478/s12175-013-0143-0