This note is about functions ƒ :
whose graph
is recognized by a Büchi finite automaton on the product alphabet . These functions are Baire class 2 in the Baire hierarchy of Borel functions
and it is decidable whether such function are continuous or not.
In 1920 W. Sierpinski showed that a function is Baire class 1 if and only if both the
overgraph and the undergraph of are . We show that
such characterization is also true for functions on infinite words
if we replace the real ordering by the...
In this paper, we study the continuity of rational functions realized by
Büchi finite state transducers. It has been shown by Prieur that it
can be decided whether such a function is continuous. We prove here that
surprisingly, it cannot be decided whether such a function has
at least one point of continuity and that its continuity set
cannot be computed. In the case of a synchronous rational function, we show that its
continuity set is rational and that it can be computed. Furthermore...
Let X be a separable Banach space and denote by 𝓛(X) (resp. 𝒦(ℂ)) the set of all bounded linear operators on X (resp. the set of all compact subsets of ℂ). We show that the maps from 𝓛(X) into 𝒦(ℂ) which assign to each element of 𝓛(X) its spectrum, approximate point spectrum, essential spectrum, Weyl essential spectrum, Browder essential spectrum, respectively, are Borel maps, where 𝓛(X) (resp. 𝒦(ℂ)) is endowed with the strong operator topology (resp. Hausdorff topology). This enables us...
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