18. A real-valued function f on a topological space S is called upper semi- continuous iff for each a ∈ R, f -1([a, ∞)) is closed, or lower semicon- tinuous iff -f is upper semicontinuous.
(a) Show that f is upper semicontinuous if and only if for all x ∈ S, f (x ) ≥ lim sup f (y) := inf{sup{ f (y): y ∈ U, y /= x }: x ∈ U open},
y→x where sup /0 := -∞.
(b) Show that f is continuous if and only if it is both upper and lower semicontinuous.
(c) If f is upper semicontinuous on a compact space S, show that for some t ∈ S, f (t ) = sup f := sup{ f (x ): x ∈ S}. Hint: Let an ∈ R, an ↑ sup f . Consider f -1((-∞, an )), n = 1, 2,... .