Primitives with respect to oscillation
The authors extend beyond metric spaces results on the existence of \(\omega\)-primitive functions. A mapping \(F:X\to\mathbb R\) is \(\omega\)-primitive to a function \(f:X\to [0,\infty)\) if the oscillation of \(F\) at every \(x\in X\) equals that of \(f(x)\) (then \(f\) is upper semicontinuous). The main result: If \(f:X\to [0,\infty)\) is upper semicontinuous and there is a dense subset of \(X\) having a dense complement that is disjoint with the set of discontinuity points of \(f\), then \(f\) admits an \(\omega\)-primitive. (Thus every continuous map \(f:X\to [0,\infty)\) on \(X\) having a dense set with dense complement admits an \(\omega\)-primitive). This is then applied to Baire dense-in-itself spaces having linearly ordered (by inclusion) bases of neighborhoods at every point.
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