  Consider the sub-Riemannian Heisenberg group $\mathbb{H}$. In this paper, we
answer the following question: given a compact set $K \subseteq \mathbb{R}$ and
a continuous map $f:K \to \mathbb{H}$, when is there a horizontal $C^m$ curve
$F:\mathbb{R} \to \mathbb{H}$ such that $F|_K = f$? Whitney originally answered
this question for real valued mappings, and Fefferman provided a complete
answer for real valued functions defined on subsets of $\mathbb{R}^n$. We also
prove a finiteness principle for $C^{m,\sqrt{\omega}}$ horizontal curves in the
Heisenberg group in the sense of Brudnyi and Shvartsman.
