solution as a function due to gradient catastrophe (shock formation) in finite time, it does have a global solution as a real multi-sheeted surface over the $(x,t)$-plane; indeed this is the construction of the method of characteristics. The sheets of this surface are obtained as the real solutions of the implicit equation $$u^\mathrm{B}=u_0(x-2u^\mathrm{B}t),
\label{eq:implicitBurgersintro}$$ and by implicit differentiation it is easy to verify that away from singularities each sheet of the surface is a function $u^\mathrm{B}=u^\mathrm{B}(x,t)$ that satisfies . A simple consequence of the Implicit Function Theorem is that for sufficiently small $|t|$ there is a unique solution of for all $x\in\mathbb{R}$. New sheets of the multivalued solution are born from *breaking points* in the $(x,t)$-plane that are in one-to-one correspondence with generic inflection points $\xi$ of $u_0$ for which $u_0'(\xi)\neq 0$ but $u_0''(\xi)=0$. If $\xi\in\mathbb{R}$ is such a point, then the corresponding breaking point is given by $$(x_\xi,t_\xi):=\left(\xi-\frac{u_0(\xi)}{u_0'(\xi)},-\frac{1}{2u_0'(\xi)}\right).$$ Each such breaking point is the location of a pitchfork bifurcation for $u^\mathrm{B}$ with respect to $t$ holding $x-2u_0(\xi)t=\xi$ fixed, with two new branches emerging as $|t|$ increases. Thus, assuming that $u_0'$ is a bounded function of total integral zero, the solution of the Cauchy problem for is classical for $$T_-:=-\frac{1}{2\max_{x\in\mathbb{R}}u_0'(x)}<t<-\frac{1}{2\min_{x\in\mathbb{R}}u_0'(x)}=:T_+.$$ 