Riesz's Rising Sun Lemma
Riesz's Rising Sun Lemma
Riesz's sunrise lemma: Let be a continuous real-valued function on such that as and as . Let there exists with . Then is an open set, and if is a finite component of , then .
f
f(x)-∞
x∞
f(x)∞
x-∞
G={x:
y>x
f(y)>f(x)}
G
(a,b)
G
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 is , and elements of are those values that remain in shadow at the instant the sun rises over the horizon as seen from .
x
f(x)
G
x
x