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Putnam
1954 Putnam
A5
A5
Part of
1954 Putnam
Problems
(1)
Putnam 1954 A5
Source: Putnam 1954
7/17/2022
Let
f
(
x
)
f(x)
f
(
x
)
be a real-valued function defined for
0
<
x
<
1.
0<x<1.
0
<
x
<
1.
If
lim
x
→
0
f
(
x
)
=
0
and
f
(
x
)
−
f
(
x
2
)
=
o
(
x
)
,
\lim_{x \to 0} f(x) =0 \;\; \text{and} \;\; f(x) - f \left( \frac{x}{2} \right) =o(x),
x
→
0
lim
f
(
x
)
=
0
and
f
(
x
)
−
f
(
2
x
)
=
o
(
x
)
,
prove that
f
(
x
)
=
o
(
x
)
,
f(x) =o(x),
f
(
x
)
=
o
(
x
)
,
where we use the O-notation.
Putnam
function
limit