Metrically separable and approximately continuous functions (Q1338193)

Following \textit{R. L. Jeffery} [The theory of functions of a real variable (1951; Zbl 0043.279)] we say that two subsets \(A\) and \(B\) of the reals \(\mathbb{R}\) are metrically separated if for arbitrary \(\varepsilon>0\), there are open sets \(O_ 1\) and \(O_ 2\) such that \(A\subset O_ 1\), \(B\subset O_ 2\) and \(m(O_ 1 \cap O_ 2)< \varepsilon\), \(m\) denotes the Lebesgue measure. Also, a function \(f: \mathbb{R}\to \mathbb{R}\) is said to be metrically separable relative to \(\mathbb{R}\) if for every real number \(c\), the sets \(\{x\in \mathbb{R}\): \(f(x)<c\}\) and \(\{x\in \mathbb{R}\): \(f(x)\geq c\}\) are metrically separated. Theorem: If \(f\) is approximately continuous almost everywhere in \(\mathbb{R}\), then it is metrically separable relative to \(\mathbb{R}\). The above theorem constitutes a converse to a result presented in Jeffery's book.