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2024 IMC
6
A function from Q to Z cannot be monotone
A function from Q to Z cannot be monotone
Source: IMC 2024, Problem 6
August 8, 2024
function
calculus
Problem Statement
Prove that for any function
f
:
Q
→
Z
f:\mathbb{Q} \to \mathbb{Z}
f
:
Q
→
Z
, there exist
a
,
b
,
c
∈
Q
a,b,c \in \mathbb{Q}
a
,
b
,
c
∈
Q
such that
a
<
b
<
c
a<b<c
a
<
b
<
c
,
f
(
b
)
≥
f
(
a
)
f(b) \ge f(a)
f
(
b
)
≥
f
(
a
)
and
f
(
b
)
≥
f
(
c
)
f(b) \ge f(c)
f
(
b
)
≥
f
(
c
)
.
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