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Miklós Schweitzer
2021 Miklós Schweitzer
3
Jensen-type condition implies continuous extension
Jensen-type condition implies continuous extension
Source: 2021 Miklos Schweitzer, P3
November 2, 2021
function
real analysis
Problem Statement
Let
I
⊂
R
I \subset \mathbb{R}
I
⊂
R
be a nonempty open interval and let
f
:
I
∩
Q
→
R
f: I \cap \mathbb{Q} \to \mathbb{R}
f
:
I
∩
Q
→
R
be a function such that for all
x
,
y
∈
I
∩
Q
x, y \in I \cap \mathbb{Q}
x
,
y
∈
I
∩
Q
,
4
f
(
3
x
+
y
4
)
+
4
f
(
x
+
3
y
4
)
≤
f
(
x
)
+
6
f
(
x
+
y
2
)
+
f
(
y
)
.
4f\left(\frac{3x + y}{4}\right)+ 4f\left(\frac{x + 3y}{4}\right) \le f(x) + 6f\left(\frac{x + y}{2}\right)+ f(y).
4
f
(
4
3
x
+
y
)
+
4
f
(
4
x
+
3
y
)
≤
f
(
x
)
+
6
f
(
2
x
+
y
)
+
f
(
y
)
.
Show that
f
f
f
can be continuously extended to
I
I
I
.
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