 
Summary: Subharmonic functions
N.A.
07
Upper semicontinuous functions. Let (X, d) be a metric space. A function
f : X R {} is upper semicontinuous (u.s.c.) if
lim inf
yx
f(y) f(x) x X.
A function g is lower semicontinuous (l.s.c.) iff g is u.s.c.
For instance, if E X, then E is u.s.c. E is closed. An increasing
function : R R is u.s.c. is rightcontinuous.
Lemma 1 f is u.s.c. f1
([a, )) is closed a R f1
([, a))
is open a R.
Proof. Exercise with sequences.
Theorem 2 (Weierstrass.) If K X is compact and f : K R {} is
u.s.c., then f has maximum (eventually, ) on K.
Proof. Let xn K be s.t. f(xn)
n
