Analysis by the method of characteristics shows that if f and $u_0$ are smooth and $u_0$ has compact support, then the Hamilton-Jacobi equation $$ (H-J) ^{u_t + f(u_x) = 0, x in R, t > 0, } _{u(x, 0) = u_0(x), x in R, } $$ has a unique $C^1$ solution u on some maximal time interval $0 leq t < T$ for which $lim_{t o T}u(x, t) exists uniformly; but this limiting function is not continuously differentiable.