  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 $f: H\to Z$ be a Baire one function. We show that there is a
continuous function $g: (X\setminus H) \to Z$ such that, for every $a\in
\partial H$, the non-tangential limit of $g$ at a equals $f(a)$ and, moreover,
if $f$ is continuous at $a\in H$ (respectively bounded in a neighborhood of
$a\in H$) then the extension $F=f\cup g$ 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.
