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Miklós Schweitzer
2015 Miklos Schweitzer
8
Functional Equation With Continuity
Functional Equation With Continuity
Source: Miklos Schweitzer
April 8, 2016
algebra
functional equation
Problem Statement
Prove that all continuous solutions of the functional equation
(
f
(
x
)
−
f
(
y
)
)
(
f
(
x
+
y
2
)
−
f
(
x
y
)
)
=
0
,
∀
x
,
y
∈
(
0
,
+
∞
)
\left(f(x)-f(y)\right)\left(f\left(\frac{x+y}{2}\right)-f\left(\sqrt{xy}\right)\right)=0 \ , \ \forall x,y\in (0,+\infty)
(
f
(
x
)
−
f
(
y
)
)
(
f
(
2
x
+
y
)
−
f
(
x
y
)
)
=
0
,
∀
x
,
y
∈
(
0
,
+
∞
)
are the constant functions.
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