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Riesz's Rising Sun Lemma

​
f(x) = α
Riesz's sunrise lemma: Let
f
be a continuous real-valued function on such that
f(x)-∞
as
x∞
and
f(x)∞
as
x-∞
. Let
G={x:
there exists
y>x
with
f(y)>f(x)}
. Then
G
is an open set, and if
(a,b)
is a finite component of
G
, then
f(a)=f(b)
.
The name of this lemma derives from the following: the sun is rising from the right in a mountainous region seen in a one-dimensional profile from the side. The elevation at
x
is
f(x)
, and elements of
G
are those
x
values that remain in shadow at the instant the sun rises over the horizon as seen from
x
.

Permanent Citation

Soledad Mª Sáez Martínez, Félix Martínez de la Rosa
​
​"Riesz's Rising Sun Lemma"​
​http://demonstrations.wolfram.com/RieszsRisingSunLemma/​
​Wolfram Demonstrations Project​
​Published: March 7, 2011
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