In this note, we will show that for a closed subanalytic subset $A \subset \mathbb{R}^n$, the Clarke tangential regularity of $A$ at $x_0 \in A$ is equivalent to the coincidence of the Clarke's tangent cone to $A$ at $x_0$ with the set \\ $$\mathcal{L}(A, x_0):= \bigg\{\dot{c}_+(0) \in \mathbb{R}^n: \, c:[0,1]\longrightarrow A\;\;\mbox{\it is Lipschitz}, \, c(0)=x_0\bigg\}.$$ Where $\dot{c}_+(0)$ denotes the right-strict derivative of $c$ at $0$. The results obtained are used to show that the Clarke regularity of the epigraph of a function may be characterized by a new formula of the Clarke subdifferential of that function.