Riesz's Rising Sun Lemma

Riesz's sunrise lemma: Let be a continuous real-valued function on ℝ such that as and as . Let there exists with . Then is an open set, and if is a finite component of , then .

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 .