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Romanian Masters of Mathematics Collection
2013 Romanian Master of Mathematics
2
2
Part of
2013 Romanian Master of Mathematics
Problems
(1)
RMM 2013 Problem 2
Source:
3/2/2013
Does there exist a pair
(
g
,
h
)
(g,h)
(
g
,
h
)
of functions
g
,
h
:
R
→
R
g,h:\mathbb{R}\rightarrow\mathbb{R}
g
,
h
:
R
→
R
such that the only function
f
:
R
→
R
f:\mathbb{R}\rightarrow\mathbb{R}
f
:
R
→
R
satisfying
f
(
g
(
x
)
)
=
g
(
f
(
x
)
)
f(g(x))=g(f(x))
f
(
g
(
x
))
=
g
(
f
(
x
))
and
f
(
h
(
x
)
)
=
h
(
f
(
x
)
)
f(h(x))=h(f(x))
f
(
h
(
x
))
=
h
(
f
(
x
))
for all
x
∈
R
x\in\mathbb{R}
x
∈
R
is identity function
f
(
x
)
≡
x
f(x)\equiv x
f
(
x
)
≡
x
?
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