If $X$ is a topological space, $Y=\mathbb{R}$, then we say that maps $g:X\rightarrow \mathbb{R}$ and $h:X\rightarrow \mathbb{R}$ form a Hahn's pair / strict Hahn's pair/, if $g$ is upper semicontinuous, $h$- lower semicontinuous and $g(x)\leq h(x)$ /$g(x)<h(x)$/ on $X$. Austrian mathematician H. Hahn proved, that every Hahn's pair $(g,h)$  on metric space $X$  has continuous intermediate function $f: X→\mathbb{R}$

For topological spaces $X$ and $Y$ we consider conditions, by which for arbitrary multivalued maps $G:X\rightarrow Y$ and $H:X\rightarrow Y$, such, that $G(x)\subseteq H(x)$ for each $x\in X$ and $G$ and $H$ are respectively upper and lower semicontinuous, there is $F:X\rightarrow Y$ continuous, such, that $G(x)\subseteq F(x)\subseteq H(x)$. We also consider conditions on topological spaces, by which the Hahn`s theorem on the intermediate function has a multivalued analog.

The set $\mathcal{P}(Y)$ is equipped with natural partial order, which is the relation of inclusion $\subseteq$ of $Y$ subsets , that allows to transfer the concept of Hahn's pair to the case of multivalued maps. We say that a multivalued maps form the Hahn /Hahn strict/ pair, if $G$ is upper semicontinuous, $H$ is lower semicontinuous and $G(x)\subseteq H(x)$ /$G(x)\subset H(x)$/ for each $x\in X$.

We show that for the normal $T_{1}$-space $X$ and any Hahn's pair $(G,H)$ of multivalued maps $G:X\rightarrow Y$ and $H:X\rightarrow Y$, which values are segments, there is intermediate continuous map $F:X\rightarrow Y$, that has segments as values in $\mathbb{R}$. Then we show, that the existence of intermediate continuous map $F:X\rightarrow Y$ for Hahn's pair $(G,H)$ of maps $G:X\rightarrow \mathbb{R}$, $G(x)=(-\infty,g(x)]$, and $H:X\rightarrow \mathbb{R}$, $H(X)=(-\infty,h(x)]$, implies that $(g,h)$ is Hahn's pair on $X$ and has continuous intermediate function $f:X\rightarrow \mathbb{R}$ on $X$.

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