Real-valued functions of a real variable which are continuous with respect to the density topology on both the domain and the range are called density continuous. A typical continuous function is nowhere density continuous. The same is true of a typical homeomorphism of the real line. A subset of the real line is the set of points of discontinuity of a density continuous function if and only if...

we characterize compact composition operators on real banach spaces of complex-valued bounded lipschitz functions on metric spaces, not necessarily compact, with lipschitz involutions and determine their spectra.

Abstract. In this paper we study the set of functions G-valued which can be approximated by G-valued continuous functions in the norm L∞G (I, w), where I is a compact interval, G is a real and separable Hilbert space and w is certain G-valued weakly measurable weight. Thus, we obtain a new extension of celebrated Weierstrass approximation theorem. Also, we characterize the set of functions whic...

For a Hausdorff zero-dimensional topological space X and totally ordered field F with interval topology, let $$C_c(X,F)$$ be the ring of all F-valued continuous functions on countable range. It is proved that if either an uncountable or subfield $${\mathbb {R}}$$ , then structure $$\beta _0X$$ Banaschewski Compactification X. The ideals $$\{O^{p,F}_c:p\in \beta _0X\}$$ in are introduced as modi...