Correspondences are continuous I want to justify if the following correspondences are continuous:


*

*$Γ$: (0,1) ⇒ $R$ defined by $Γx=[0,1/n]$;

*$Γ: R_+⇒R$ defined by $Γx=[0,1]$ for $x≠1$ and $Γ1=(0,2]$.


I know that a correspondence is continuous if it is lower semi-continuous and upper semi-continuous. I also know the following definitions:
The correspondence is $Γ: X⇒Y$ is lower semi-continuous at $x$ if for each $G⊂Y$, $G∈ τ_Y $, $G ∩ τ_x ≠∅$, there is some $U∈N_x$ such that $y∈U⇒Γ_y∩G≠0$. The corresponde is lower semi-continuous if it is so at each $x∈X$.
The correspondence is $Γ: X⇒Y$ is upper semi-continuous at $x$ if for each $G⊂Y$, $G∈ τ_Y $, $τ_x⊂G$, there is some $U∈N_x$ such that $y∈U⇒Γ_y⊂G$. The corresponde is upper semi-continuous if it is so at each $x∈X$.
My first guess is that the correspondence in example 1 is continuous and in the 2nd example is not. My question is: how can I use this definitions to justify if the correspondences in the examples are continuous or not?
 A: I think you meant to define upper hemi-continuity and lower hemi-continuity, since these are correspondences and not functions. I am also going to guess that you meant to write $\Gamma(x)=[0,x]$ in the first case.
$\Gamma: A \rightarrow B$ is $upper~hemi-continuous$ at $a$  if for every sequence $a_n \rightarrow a$ and every sequence with $b_n \in \Gamma(a_n)$ with $b_n \rightarrow b$ it is true that $b \in \Gamma(a)$.
$\Gamma: A \rightarrow B$ is $lower~hemi-continuous$ at $a$  if for every sequence $a_n \rightarrow a$ there exists a corresponding sequence with $b_n \in \Gamma(a_n)$ such that  $b_n \rightarrow b$.
A correspondence is continuous at $a$ if it satisfies both of these properties.  It is continuous if it satisfies these properties at every $a$ in its domain.
The first correspondence is upper hemi-continuous because for every sequence $x_n \rightarrow a \in (0,1)$ and every $b_n\in \Gamma(x_n)=[0,x_n]$ with $b_n \rightarrow b$, $b \in \Gamma(a)=[0,a]$.  (If $b \notin [0,a]$, then for every $N$ there would be $n \ge N$ with $b_n \notin \Gamma(a_n)$, contradiciting our assumption.)
The first correspondence is lower hemi-continuous.  The only problem might occur for a  sequence $x_n \rightarrow 0$ with $x_n > 0$  But there is always a sequence 
$$
z_n \in \Gamma(x_n)=[0,x_n] \\
z_n \rightarrow 0
$$ 
So the first correspondence is continuous.
The second correspondence is not upper hemi-continuous because
$$
x_n=1-1/n\\
x_n \rightarrow 1\\
0 \in \Gamma(x_n) \\
0 \notin \Gamma(1) = (0, 2]
$$ 
