## Abstract

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 f: H→ Z be a Baire one function. We show that there is a continuous function g: ( X \ H)→ Z such that, for every a∈ ∂H, the non-tangential limit of g at a equals f(a) and, moreover, if f is continuous at a∈ H (respectively bounded in a neighborhood of a∈ H) then the extension F= f∪g 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. In an accompanying paper (Extensions of vector-valued functions with preservation of derivatives), the main result is applied to extensions of vector-valued functions defined on a closed subset of Euclidean or Banach space with preservation of differentiability, continuity and (pointwise) Lipschitz property.

Original language | English |
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Pages (from-to) | 138-148 |

Number of pages | 11 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 442 |

Issue number | 1 |

Early online date | 20 Apr 2016 |

DOIs | |

Publication status | Published - 1 Oct 2016 |

## Keywords

- Continuity points
- Continuous convergence
- Extensions
- Non-tangential limit
- Pointwise approximation
- Vector-valued Baire one functions

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics