Smooth extension of a function

Let $U\subseteq V\subseteq\mathbb R^n$ be open and $f:U\to\mathbb R^m$ smooth. A smooth extension of $f$ to $V$ is a smooth function $F:V\to\mathbb R^m$ whose restriction to $U$ equals $f$ .

Existence and nonuniqueness

An extension need not exist: $1/t$ on $(0,1)$ has no continuous extension across zero. When one exists it need not be unique, since smooth functions supported in $V\setminus\overline U$ may be added without changing the restriction.

The zero-extension criterion gives a useful sufficient condition across a flat boundary. Borel's extension theorem concerns prescribing boundary derivatives; it does not assert that an arbitrary function on an open domain extends smoothly.