The relations between upper continuity, lower continuity, $H^+$-continuity and   $H^-$-continuity for compact-valued and closed-valued mappings are investigated. Besides, we construct a closed-valued $H$-continuous mappings $F: \mathbb{R} → \mathbb{R}$ which are not everywhere upper continuous but are pointwise H-limits of continuous and $H$-continuous closed-valued mappings $F_n: \mathbb{R} → \mathbb{R}$ in Hausdorff metric or in Vietoris topology.

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