I am studying the definition of upper hemicontinuity. The definition says a correspondence $F:X\rightrightarrows Y$ is said to be upper hemicontinuous at $x$, if, for every open subset $O\subseteq Y$ with $F(x)\subseteq O$, there exists a $\delta>0$ such that $$F(N_{\delta,X}(x))\subseteq O.$$
In the example I attached, there is an open set $O=(y_2-\varepsilon,y_1+\varepsilon)\cup (y_4-\varepsilon,y_3+\varepsilon)$ for some $\varepsilon>0$, which contains $F(x)$. Then there is $\delta>0$ (in fact, all $\delta>0$) such that $F(x')\subset O$ for any $x'$ in the neighborhood of $x$ with $\delta$.
I believe this should be wrong. Otherwise, Kakutani's fixed point theorem does not go through. Can you help me to correct my understanding about upper hemicontinuity?
The (middle) line $f(x)=x$ is not part of correspondence $F$. I drew $f(x)=x$ to show that Kakutani's fixed point theorem doesn't hold in $F$. Thanks!!