Lebesgue sets and insertion of a continuous function

Lane, Ernest P.

doi:10.1090/S0002-9939-1983-0684654-0

Lebesgue sets and insertion of a continuous function
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Proc. Amer. Math. Soc. 87 (1983), 539-542 Request permission

Abstract:

Necessary and sufficient conditions in terms of Lebesgue sets are presented for the following two insertion properties for real-valued functions defined on a topological space: (1) $g \leqslant f$ there is a continuous function $h$ such that $g \leqslant h \leqslant f$, and for each $x$ for which $g(x) < f(x)$ then $g(x) < h(x) < f(x)$. (2) $g < f$ there is a continuous function $h$ such that $g < h < f$.

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