  The notion of strict singular characteristics is important in the
wellposedness issue of singular dynamics on the cut locus of the viscosity
solutions. We provide an intuitive and rigorous proof of the existence of the
strict singular characteristics of Hamilton-Jacobi equation $H(x,Du(x),u(x))=0$
in two dimensional case. We also proved if $\mathbf{x}$ is a strict singular
characteristic, then we really have the right-differentiability of $\mathbf{x}$
and the right-continuity of $\dot{\mathbf{x}}^+(t)$ for every $t$. Such a
strict singular characteristic must give a selection $p(t)\in
D^+u(\mathbf{x}(t))$ such that $p(t)=\arg\min_{p\in
D^+u(\mathbf{x}(t))}H(\mathbf{x}(t),p,u(\mathbf{x}(t)))$.
