ABSTRACT. The notion of porouscontinuous function will be introduced on the base of porous set and relations between porouscontinuous, continuous and quasicontinuous functions will be investigated.
The notion of quasicontinuity was perhaps the first time used by R. Baire in [2]. Let X, Y be topological spaces and Q(X, Y) be the space of quasicontinuous mappings from X to Y. If X is a Baire space and Y is metrizable, in Q(X, Y) we can approach each (x, y) in the graph Grf of f along some trajectory of the form {(x k , f n k (x k)) : k ∈ ω} if and only if we can approach most points along a vertical trajectory. This result generalizes Theorem 5 from [3]. Moreover in the class of topological spaces with the property QP we give a characterization of Baire spaces by the above mentioned fact. We also study topological spaces with the property QP.
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Let (X, g) be a generalized topological space, (Y, d) a metric one and f : X → Y a function. We can define a generalized oscillation of f at x ∊ X as kgf (x) = inf{diamf(A) : A ∊ g, x ∊ A}. We discuss some properties of the generalized oscillation.
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