Problem of the existence of ω*-primitives

lf (X, ᵨ) is a dense in itself metric space and f : X →ℝ, then we define ω*(f,x) = infr >0 supy,z∈B (x,r) \ {x} ׀ f(y) - f(z)׀. We say that a function F : X →ℝ is an ω*-primitive for f : X →ℝ if ω* (F, .) = f. We discuss problem of the existence of ω*-primitives for an arbitrary upper semicontinuous function f : X → [0, ∞ ) defined on a dense in itself metric space. At the end we show that if an upper semicontinuous function f : X → [0, ∞) is defined on a nonmetrizable topological space, then ω*-primitive may not exists.