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District Olympiad
2003 District Olympiad
2
Romania District Olympiad 2003 - Grade XI
Romania District Olympiad 2003 - Grade XI
Source:
March 18, 2011
function
real analysis
real analysis unsolved
Problem Statement
Let
f
:
[
0
,
1
]
→
[
0
,
1
]
f:[0,1]\rightarrow [0,1]
f
:
[
0
,
1
]
→
[
0
,
1
]
a continuous function in
0
0
0
and in
1
1
1
, which has one-side limits in any point and
f
(
x
−
0
)
≤
f
(
x
)
≤
f
(
x
+
0
)
,
(
∀
)
x
∈
(
0
,
1
)
f(x-0)\le f(x)\le f(x+0),\ (\forall)x\in (0,1)
f
(
x
−
0
)
≤
f
(
x
)
≤
f
(
x
+
0
)
,
(
∀
)
x
∈
(
0
,
1
)
. Prove that:a)for the set
A
=
{
x
∈
[
0
,
1
]
∣
f
(
x
)
≥
x
}
A=\{x\in [0,1]\ |\ f(x)\ge x\}
A
=
{
x
∈
[
0
,
1
]
∣
f
(
x
)
≥
x
}
, we have
sup
A
∈
A
\sup A\in A
sup
A
∈
A
. b)there is
x
0
∈
[
0
,
1
]
x_0\in [0,1]
x
0
∈
[
0
,
1
]
such that
f
(
x
0
)
=
x
0
f(x_0)=x_0
f
(
x
0
)
=
x
0
.Mihai Piticari
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