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TA: Yankun Wang
Econ 617 Problem Set 1
Solution Key
1.
The whole point of this problem is to understand the de
f
tion of continuity and to get
used to the
ε
−
δ
argument.
(a): For
f
:
R
+
→
R
+
,
we say that
f
is continuous if: for every
x
∗
∈
R
+
and for
any
ε>
0
,w
ecan
f
nd a
δ>
0
,
and
δ
depending on
ε,
such that for all
x
satisfying

x
−
x
∗

<δ
and
x
∈
R
+
,wehave

f
(
x
)
−
f
(
x
∗
)

<ε.
(i)
f
(
x
)=
x
:
pick an
arbitrary
x
∗
∈
R
+
,
and for any
ε>
0
,
let
δ
=
ε
:
the rest should
be obvious.
(ii)
f
(
x
)=1+
x
:
again, pick an
arbitrary
x
∗
∈
R
+
,
and let
δ
=
ε
:
for all
x
satisfying

x
−
x
∗

<δ
and
x
∈
R
+
,

f
(
x
)
−
f
(
x
∗
)

=

x
−
x
∗

<δ
=
ε.
(iii)
f
(
x
)=
1
1+
x
:
pick an
arbitrary
x
∗
∈
R
+
,
and let
ε>
0
.F
o
ra
n
y
x
∈
R
+
,

f
(
x
)
−
f
(
x
∗
)

=

1
1+
x
−
1
1+
x
∗

=

x
∗
−
x
(1 +
x
)(1 +
x
∗
)

=

x
∗
−
x

(1 +
x
)(1 +
x
∗
)
.
Since
1+
x
≥
1
,
1+
x
∗
≥
1
,
we have
0
<
1
(1+
x
)(1+
x
∗
)
≤
1
.
Now just let
δ
=
ε,
for
x
satisfying

x
−
x
∗

<δ
and
x
∈
R
+
,

f
(
x
)
−
f
(
x
∗
)

<δ
=
ε.
To draw of the graph of this function, notice that we can obtain it by shift the graph
of
f
(
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This note was uploaded on 08/29/2009 for the course ECON 617 taught by Professor Staff during the Fall '08 term at Cornell University (Engineering School).
 Fall '08
 STAFF

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