Summary: The \(k\)-th order tangent star, \(\text{Tan}^ k(M,X)\), of a closed subset \(X\) of a \(C^ k\)-manifold \(M\) is defined and studied. The map \((M,X)\mapsto \text{Tan}^ k(M,X)\) is a covariant functor from the category of pairs to the category of stars. Given a continuous function \(f: X\to\mathbb{R}\), and letting \(G=\text{graph }f\), we consider the star- morphism \[ \pi_ *:\text{Tan}^ k(M\times\mathbb{R},G)\to\text{Tan}^ k(M,X) \] induced by the projection \(\pi: M\times\mathbb{R}\to M\). Theorem: The function \(f\) has a \(C^ k\)-extension to \(M\) if and only if \(\pi_ *\) is a bijection. A method for calculating \(\text{Tan}^ k(M,X)\), and several examples, are presented, and the relations to other tangent concepts are investigated.