Continuity of Correspondences

Question

$\varphi(\theta)$ : $\mathbb{R} \Rightarrow \mathbb{R}$ be a correspondence. Show whether $\varphi(\theta)$ below is upper hemicontinuous or(and) lower hemiconinuous.

a)

$\varphi(\theta)$ = $ \begin{cases} [-1, 1] & for & \theta=0 \\ {sin {1\over s}} & for & \theta \neq0 \end{cases}$

b)

$\varphi(\theta)$ = $ \begin{cases} (-1, 2) & for & \theta=0 \\ {sin {1\over s}} & for & \theta \neq0 \end{cases}$

Should I go through the definition; a correspondence $\varphi: \oplus \rightarrow P(S)$ is said to be upper hemicontinuous at point $\theta \in \oplus$ if for all open sets V such that $\varphi(\theta)\subset V$, there exist an open set U containing $ \theta $ , such that $ \theta^´ \in U \cap \oplus $ implies $\varphi(\theta^´) \subset V$. We say that $\varphi$ is upper hemicontinuous on $\oplus$ if $\varphi$ is upper hemicontinuous at each $\theta \in \oplus$.

Actually, I am kind of confused with this definition. Are there any other methods that I can use to obtain continuity analysis of correspondences ? What should be the first step to take in these kind of questions ?

I would be glad if you help with these questions. Thank you.

Answer 1

You might find the following definitions helpful. Let $(X,d_1)$ and $(Y,d_2)$ be metric spaces. A non-empty valued correspondence $\Gamma : X \rightarrow Y$ is lower hemicontinuous at $x\in X$ if for all $\{x_n\}_{n=0}^\infty \subset X$ s.t. $x_n\rightarrow x$ and for all $y\in\Gamma(x)$ we can find $\{y_n\}_{n=0}^\infty\subset Y$ such that $y_n\rightarrow y$ and $y_n\in\Gamma(x_n)$ for all $n\in\mathbb{N}$. The correspondence is lower hemicontinuous if it is lower hemicontinuous at all $x\in X$. A non-empty, compact valued correspondence $\Gamma : X \rightarrow Y$ is upper hemicontinuous at $x\in X$ if for every sequence $\{x_n\}_{n=0}^\infty\subset X$ s.t $x_n\rightarrow x$ every sequence $\{y_n\}_{n=0}^\infty\subset Y$ s.t. $y_n\in \Gamma(x_n)$ for all $n\in\mathbb{N}$ we can find a convergent subsequence of $y_n$ s.t. it converges to a point $y\in\Gamma(x)$. The correspondence is upper hemicontinuous if it is upper hemicontinuous at all $x\in X$. The correspondence is continuous if it is both l.h.c and u.h.c.