Returning functions with closed graph are continuous

Year, pages: 2020, 297 - 304

A function $f:X\to \IR$ defined on a topological space $X$ is called {\em returning} if for any point $x\in X$ there exists a positive real number $M_x$ such that for every path-connected subset $C_x\subset X$ containing the point $x$ and any $y\in C_x\smallsetminus\{x\}$ there exists a point $z\in C_x\smallsetminus\{x,y\}$ such that $|f(z)|≤ \max\{M_x,|f(y)|\}$. A topological space $X$ is called {\em path-inductive} if a subset $U\subset X$ is open if and only if for any path $γ:[0,1]\to X$ the preimage $γ^-1(U)$ is open in $[0,1]$. The class of path-inductive spaces includes all first-countable locally path-connected spaces and all sequential locally contractible spaces.

Banakh, T., Filipczak, M., Wódka , J. 2020. Returning functions with closed graph are continuous. In Mathematica Slovaca, vol. 70, no.2, pp. 297-304. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0352

Banakh, T., Filipczak, M., Wódka , J. (2020). Returning functions with closed graph are continuous. Mathematica Slovaca, 70(2), 297-304. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0352

Publisher: Mathematical Institute, Slovak Academy of Sciences, Bratislava