On chain recurrent sets of typical bounded Baire one functions

Uniform limits of $\mathcal B_1^{**}$-functions

“…A different kind of application reveals a work by one of the authors [10], where the B * 1 property was used as a main tool for a characterisation of integrability analogous to the Lebesgue theorem on the Riemann integral. 1 Under the Baire-one-star name, and in generalised differentiation connection, the property was introduced by Richard J. O'Malley in [11], but is also known as the Baire half property after Harvey Rosen's paper [13], or as piecewise continuity [3]. (See [6] for more info, various descriptions of B * 1 and related properties.)…”

“…(E s+1 \ int E s ) ,is a continuous function and its domain,(1), is a closed subset of X \ D. Thus, from the normality of X \ D, there is a continuous extension g of lim k→∞ g k defined on X \ D. Set f 2 as g on X \ D and as 0 on D. Clearly,f 2 ∈ B * * 1 ⊂ uS 1 . It remains to check that f 3 = f − f 1 − f 2 is a uS n function.…”

“…Key words and phrases. Baire-one-star function, Pawlak's class B * * 1 , Małek's classes Sα, classes uSα, oscillation rank, Katětov-Tong insertion theorem.1 Quite recently, B * 1 functions received some attention in connection with dynamical systems, as so-called k-continuous functions or right B 1 compositors[1].…”