EDB — 13D

[13D]Let \(f:X→ ℝ\); the following assertions are equivalent.

1. \(f\) is lower semicontinuous.

2. For every \(t\), we have that the sublevel

   \[ S_ t = \{ x∈ X, f(x)≤ t \} \]

   is closed.

3. The epigraph

   \[ E = \{ (x,t)∈ X×ℝ, f(x)≤ t \} \]

   is closed in \(X× ℝ\).

Note that the second condition means that \(f\) is continuous from \((X,𝜏)\) to \(ℝ,𝜏_+\) where \(𝜏_+=\{ (a,∞):a∈ℝ\} ∪\{ ∅,ℝ\} \) is the set of half-lines, which is a topology (easy verification).

Then formulate the equivalent theorem for functions upper semicontinuous.