Let $\phi: X \rightrightarrows Y$ be an upper hemicontinuous correspondence. If $K \subset X$ is a compact and convex set, and $\phi(x)$ is nonempty, compact, and convex for any $x \in K$, and $\bigcup_{x \in K} \phi(x)$ is also bounded, then is there a point $x_0 \in K$ such that $\phi$ is continuous at $x_0$?

Since analogous results hold for real-valued lower semicontinuous functions, I would expect the above statement to be true. But I haven't found any references.

Any help is appreciated. Thanks a lot!

Let $\phi: X \rightrightarrows Y$ be an upper hemicontinuous correspondence. If $K \subset X$ is a compact and convex set, $K$ contains an open set $U$, and $\phi(x)$ is nonempty, compact, and convex for any $x \in K$, and $\bigcup_{x \in K} \phi(x)$ is also bounded, then is there a point $x_0 \in K$ such that $\phi$ is continuous at $x_0$?

Since analogous results hold for real-valued lower semicontinuous functions, I would expect the above statement to be true. But I haven't found any references.

Any help is appreciated. Thanks a lot!