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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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