Extensions of vector-valued Baire one functions with preservation of points of continuity

0459546 - MÚ 2017 RIV US eng J - Journal Article
Koc, M. - Kolář, Jan
Extensions of vector-valued Baire one functions with preservation of points of continuity.
Journal of Mathematical Analysis and Applications. Roč. 442, č. 1 (2016), s. 138-148. ISSN 0022-247X. E-ISSN 1096-0813
R&D Projects: GA ČR(CZ) GA14-07880S
Institutional support: RVO:67985840
Keywords : vector-valued Baire one functions * extensions * non-tangential limit * continuity points
Subject RIV: BA - General Mathematics
Impact factor: 1.064, year: 2016
http://www.sciencedirect.com/science/article/pii/S0022247X1630097X

We prove an extension theorem (with non-tangential limits) for vector-valued Baire one functions. Moreover, at every point where the function is continuous (or bounded), the continuity (or boundedness) is preserved. More precisely: Let H be a closed subset of a metric space X and let Z be a normed vector space. Let ... be a Baire one function. We show that there is a continuous function ... such that, for every ..., the non-tangential limit of g at a equals f(a)f(a) and, moreover, if f is continuous at ... (respectively bounded in a neighborhood of ...) then the extension ... is continuous at a (respectively bounded in a neighborhood of a). We also prove a result on pointwise approximation of vector-valued Baire one functions by a sequence of locally Lipschitz functions that converges “uniformly” (or, “continuously”) at points where the approximated function is continuous.
Permanent Link: http://hdl.handle.net/11104/0259733