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Abstract

Let $(X,{\mathcal T})$ be any $T_1$ topological space. Given a function $F\colon X\to\mathbb{R}$ and $x\in X$, we define the oscillation of $F$ at $x$ to be $\omega(F,x)=\inf_{U}\sup_{x_1,x_2\in U}|F(x_1)-F(x_2)|$, where the infimum is taken over all neighborhoods $U$ of $x$. It is well known that $\omega(F,\cdot)\colon X\to [0,\infty]$ is upper semicontinuous and vanishes at all isolated points of $X$.

Suppose an upper semicontinuous function $f\colon X\to [0,\infty]$ vanishing at isolated points of $X$ is given. If there exists a function $F\colon X\to \mathbb{R}$ such that $\omega(F,\cdot)=f$, then we call $F$ an $\omega$-primitive for $f$. By the `$\omega$-problem' on a topological space $X$ we mean the problem of the existence of an $\omega$-primitive for a given upper semicontinuous function vanishing at all isolated points of $X$.