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CIIM
2015 CIIM
Problem 4
CIIM 2015 Problem 4
CIIM 2015 Problem 4
Source:
August 9, 2016
CIIM 2015
undergraduate
Problem Statement
Let
f
:
R
→
R
f:\mathbb{R} \to \mathbb{R}
f
:
R
→
R
a continuos function and
α
\alpha
α
a real number such that
lim
x
→
∞
f
(
x
)
=
lim
x
→
−
∞
f
(
x
)
=
α
.
\lim_{x\to\infty}f(x) = \lim_{x\to-\infty}f(x) = \alpha.
x
→
∞
lim
f
(
x
)
=
x
→
−
∞
lim
f
(
x
)
=
α
.
Prove that for any
r
>
0
,
r > 0,
r
>
0
,
there exists
x
,
y
∈
R
x,y \in \mathbb{R}
x
,
y
∈
R
such that
y
−
x
=
r
y-x = r
y
−
x
=
r
and
f
(
x
)
=
f
(
y
)
.
f(x) = f(y).
f
(
x
)
=
f
(
y
)
.
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