سمیرا رهروی

In this thesis, we have investigated joint upper and lower semicontinuity of twovariable set-valued functions. More precisely, among other results, we shown have that under certain conditions, a two-variable lower horizontally quasicontinuous mapping F : X × Y → K(Z) is jointly upper semicontinuous on sets of the from D × {y0}, where D is a dense Gδ subset of X and y0 ∈ Y . Similar result for joint lower semicontinuity of upper horizontally quasicontinuous mappings is obtained. These have improved some known results on joint continuity of singlevalued functions. Also, Let X be a Baire space, Y be a compact Hausdorff space and φ : X −→ Cp(Y ) be a quasicontinuous mapping. For a proximal subset H of Y × Y we have used topological games G1(H) and G2(H) on Y × Y between two players to prove that if the first player has a winning strategy in these games, then φ is norm continuous on a dense Gδ subset of X. It follows that if Y is Valdivia compact, each quasicontinuous mapping from a Baire space X to Cp(Y ) is norm continuous on a dense Gδ subset of X. also, we prove that a compactvalued multifunction F : X × Y −→ Z, where X is a Baire space and Y , Z are separable metrizable spaces, is quasi-continuous if and only if F is horizontally quasi-continuous and there exists an residual subset M of X such that for any x ∈ M the multifunction Fx = F(x, ·) is quasi-continuous on Y .