Atsushi Inoue
ECG 765: Mathematical Methods for Economics
Fall 2009
Lecture Notes 4
Continuous Functions and Existence of Optimal Solutions
We will present sufficient conditions for an optimization problem to have a
solution.
Outline
A. Continuous Functions
B. Existence of Optimal Solutions
A. Continuous Functions
Definition.
Let
x
0
be a point in the domain
X
of the function
f
. We say that
f
is
continuous at
x
0
if, for a given
>
0, there exists a
δ
( )
>
0 such that if
x
∈
X
and
d
(
x, x
0
) =
k
x

x
0
k
< δ
( ), then
d
(
f
(
x
)
, f
(
x
0
)) =

f
(
x
)

f
(
x
0
)

<
. If
f
is continuous at every point of
X
, we say that
f
is
continuous on
X
.
Examples.
f
(
x
)
=
a
+
bx.
g
(
x
)
=
(
0
if
x < z
1
if
x
≥
z
Definition.
(a) If
x
is a point in
S
, then any set which contains an open set containing
x
is called a
neighborhood
of
x
.
(b) A point
x
0
is a
limit point
(
accumulation point
) if
every
neighborhood
of
x
0
contains at least one point of
S
distinct from
x
0
.
1
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Example.
S
= (
∞
,

1)
∪ {
0
} ∪
(1
,
∞
).
Definition.
Let
f
:
X
→ <
and
x
0
be a limit point of
X
. We write
f
(
x
)
→
y
0
as
x
→
x
0
, or
lim
x
→
x
0
f
(
x
) =
y
0
,
if there is a point
y
0
∈ <
with the following property: For every
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 Fall '07
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 Topology, Continuous function, Atsushi Inoue, A. Continuous Functions

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