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Let $X, Y$ be topological spaces. A function $f: X \to Y$ is said to be graph quasicontinuous if there is a quasicontinuous function $g: X \to Y$ with the graph of $g$ contained in the closure of the graph of $f$. There is a close relation between the notions of graph quasicontinuous functions and minimal usco maps as well as the notions of graph quasicontinuous functions and densely continuous forms. Every function with values in a compact Hausdorff space is graph quasicontinuous; more generally every locally compact function is graph quasicontinuous.
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