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تسجيل مستخدم جديدReturning functions with closed graph are continuous
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A function $f:Xto mathbb R$ defined on a topological space $X$ is called returning if for any point $xin X$ there exists a positive real number $M_x$ such that for every path-connected subset $C_xsubset X$ containing the point $x$ and any $yin C_xsetminus{x}$ there exists a point $zin C_xsetminus{x,y}$ such that $|f(z)|le max{M_x,|f(y)|}$. A topological space $X$ is called path-inductive if a subset $Usubset X$ is open if and only if for any path $gamma:[0,1]to X$ the preimage $gamma^{-1}(U)$ is open in $[0,1]$. The class of path-inductive spaces includes all first-countable locally path-connected spaces and all sequential locally contractible space. We prove that a function $f:Xto mathbb R$ defined on a path-inductive space $X$ is continuous if and only of it is returning and has closed graph. This implies that a (weakly) Swic atkowski function $f:mathbb Rtomathbb R$ is continuous if and only if it has closed graph, which answers a problem of Maliszewski, inscibed to Lviv Scottish Book.
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