Suppose there exists a smooth function $f$ from $U \subset R^n \rightarrow R$ (smooth at every point in $U$). Is it true that there exist a function $g : V \supset U \rightarrow R$ in an open set $V$ containing $U$ such that g is smooth $V$?
Definition of “smooth at a point”
For any set $A \subset R^n$, and $a \in A$, we say that a function $ f : A \rightarrow R$ is smooth at a if there is a smooth function $g: U \rightarrow R$ defined in a neighbourhood of a such that $g=f$ on $U \cap S$.