continuous function attains its minimum on a compact set. Such statement is
the simplest version of the Weierstrass theorem. In this section we prove an
extended version. We need some preliminary results and definitions.
Proposition
(Closeness and lower
semicontinuity). Let
be a function
.
The following statements are equivalent:
1. The level sets
are closed for every
.
2. The function
is lower semicontinuous.
3. The
is a closed set.
