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2003 IMC
5
IMC 2003 Problem 11
IMC 2003 Problem 11
Source: IMC 2003 Day 2 Problem 5
November 2, 2020
function
college contests
real analysis
IMC
Problem Statement
a) Show that for each function
f
:
Q
×
Q
→
R
f:\mathbb{Q} \times \mathbb{Q} \rightarrow \mathbb{R}
f
:
Q
×
Q
→
R
, there exists a function
g
:
Q
→
R
g:\mathbb{Q}\rightarrow \mathbb{R}
g
:
Q
→
R
with
f
(
x
,
y
)
≤
g
(
x
)
+
g
(
y
)
f(x,y) \leq g(x)+g(y)
f
(
x
,
y
)
≤
g
(
x
)
+
g
(
y
)
for all
x
,
y
∈
Q
x,y\in \mathbb{Q}
x
,
y
∈
Q
. b) Find a function
f
:
R
×
R
→
R
f:\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}
f
:
R
×
R
→
R
, for which there is no function
g
:
Q
→
R
g:\mathbb{Q}\rightarrow \mathbb{R}
g
:
Q
→
R
such that
f
(
x
,
y
)
≤
g
(
x
)
+
g
(
y
)
f(x,y) \leq g(x)+g(y)
f
(
x
,
y
)
≤
g
(
x
)
+
g
(
y
)
for all
x
,
y
∈
R
x,y\in \mathbb{R}
x
,
y
∈
R
.
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