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ISI B.Math Entrance Exam
2008 ISI B.Math Entrance Exam
1
B.Math 2008-Integration .
B.Math 2008-Integration .
Source: 10+2
April 16, 2012
calculus
integration
function
calculus computations
Problem Statement
Let
f
:
R
→
R
f:\mathbb{R} \to \mathbb{R}
f
:
R
→
R
be a continuous function . Suppose
f
(
x
)
=
1
t
∫
0
t
(
f
(
x
+
y
)
−
f
(
y
)
)
d
y
f(x)=\frac{1}{t} \int^t_0 (f(x+y)-f(y))\,dy
f
(
x
)
=
t
1
∫
0
t
(
f
(
x
+
y
)
−
f
(
y
))
d
y
∀
x
∈
R
\forall x\in \mathbb{R}
∀
x
∈
R
and all
t
>
0
t>0
t
>
0
. Then show that there exists a constant
c
c
c
such that
f
(
x
)
=
c
x
∀
x
f(x)=cx\ \forall x
f
(
x
)
=
c
x
∀
x
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