Let $(X,d)$ and $(Y,d)$ be complete metric space, and denote by $H(Y,d)$ the Hausdorff metric space on the compact subsets of $(Y,d)$. Let $$ f,g:(X,d)\rightarrow H(y,d), $$ be continuous maps (may be assumed to be $\epsilon$-Lipschitz, for $\epsilon \in [0,1]$ if necessary). Then define the function $h$ by $$ \begin{aligned} h: &(X,d) \rightarrow H(y,d)\\ h&(x)\to f(x)\cap g(x); \end{aligned} $$ assume that $h(x)\neq \emptyset$ for every $x \in X$.
Then must $h$ be continuous?
Let X=Y=[0,1], and d be the Euclidean metric. Let f(x)={0,x} and g(x)={0,1-x}, then f and g are continuous. h(x) is not empty for all x in X, consider a sequence converges to 1/2, h is not contiunous at 1/2.