A correspondence is upper (lower) hemicontinuous if it is upper (lower) hemicontinuous at all
x
∈
X
. It is continuous if it is both upper and lower hemicontinuous
It is often easier to check sufficient conditions for a correspondence’s continuity that to verify
the basic definition directly. Here are two sometimes convenient sets of conditions.
Proposition 5.11.
Suppose
X
⊆
R
m
and
Y
⊆
R
n
. A compact-valued correspondence
f
:
X
→
Y
is upper hemicontinuous if, and only if, for any domain sequence
x
j
→
x
and any range sequence
y
j
such that
y
j
∈
f
(
x
j
)
, there exists a convergent subsequence
{
y
j
k
}
such that
lim
y
j
k
∈
f
(
x
)
.
Proposition 5.12.
Suppose
A
⊆
R
m
and
B
∈
R
n
is closed. If
f
:
A
⇒
B
has closed graph and
f
(
K
)
is bounded for any compact
K
⊆
A
, then
f
is upper hemicontinous.
10
To prove that the graph of Γ :
A
⇒
B
is closed, show that the graph contains all of its limit
points: if (
x
n
, y
n
)
→
(
x, y
) and
y
n
∈
Γ(
x
n
) for all
n
(i.e. each (
x
n
, y
n
) is in the graph of Γ), then
y
∈
Γ(
x
) (i.e. the limit (
x, y
) is in the graph of Γ).
Proposition 5.13.
Suppose
A
⊆
R
m
,
B
⊆
R
n
, and
f
:
A
⇒
B
. Then
f
is lower hemicontinuous
if, and only if, for all
{
x
m
} ∈
A
such that
x
m
→
x
∈
A
and
y
∈
f
(
x
)
, there exist
y
m
∈
f
(
x
m
)
such
that
y
m
→
y
.
Exercise 5.14.
Suppose Γ :
R
⇒
R
is defined by:
Γ(
x
) =
(
{
1
}
if
x <
1
[0
,
2]
if
x
≥
1
.
10
The
graph
of
f
is the subset
G
⊆
A
×
B
defined by (
a, b
)
∈
G
if
b
∈
f
(
a
). The image
f
(
K
) of a set
K
is defined
as
f
(
K
) =
S
x
∈
K
f
(
x
).
25

Prove that Γ is upper hemicontinuous, but not lower hemicontinuous.
Suppose Γ
0
:
R
⇒
R
is
defined by:
Γ
0
(
x
) =
(
{
1
}
if
x
≤
1
[0
,
2]
if
x >
1
.
Prove that Γ
0
is lower hemicontinuous, but not upper hemicontinuous.

Exercise 5.15.
Suppose
g
:
A
→
B
is a function. Define Γ :
A
⇒
B
by Γ(
x
) =
{
g
(
x
)
}
. Prove that
g
is a continuous function if and only if Γ is a continuous correspondence.