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Let (X,) be any T₁ topological space. Given a function F: X → ℝ and x ∈ X, we define the oscillation of F at x to be $\omega (F,x)=in{f}_{U}su{p}_{x₁,x₂\in U}|F\left(x₁\right)-F\left(x₂\right)|$, where the infimum is taken over all neighborhoods U of x. It is well known that ω(F,·): X → [0,∞] is upper semicontinuous and vanishes at all isolated points of X. Suppose an upper semicontinuous function f: X → [0,∞] vanishing at isolated points of X is given. If there exists a function F: X → ℝ such that ω(F,·)=f, then we call F an ω-primitive for f. By the ’ω-problem’ on a topological...