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

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→Zf:H→Z be a Baire one function. We show that there is a continuous function g:(X∖H)→Zg:(X∖H)→Z such that, for every a∈∂Ha∈∂H, the non-tangential limit of g at a   equals f(a)f(a) and, moreover, if f   is continuous at a∈Ha∈H (respectively bounded in a neighborhood of a∈Ha∈H) then the extension F=f∪gF=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.