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Sweden Contests
Swedish Mathematical Competition
1963 Swedish Mathematical Competition.
4
4
Part of
1963 Swedish Mathematical Competition.
Problems
(1)
f(x+y) = f(x) + f(y) + f(kxy) , differentiable
Source: 1963 Swedish Mathematical Competition p4
3/21/2021
Given the real number
k
k
k
, find all differentiable real-valued functions
f
(
x
)
f(x)
f
(
x
)
defined on the reals such that
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
+
f
(
k
x
y
)
f(x+y) = f(x) + f(y) + f(kxy)
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
+
f
(
k
x
y
)
for all
x
,
y
x, y
x
,
y
.
functional equation
functional
algebra
analysis