Prove that a DE solution does not exist after a certain time I am unable to begin/know what do in order to show the following:


  Consider the one dimensional system
  $$
\dot x(t)=-\text{sgn}(x(t))=\begin{cases}
-1 &amp; \text{if } x(t)\ge 0 \\
1 &amp; \text{if } x(t)&lt;0
\end{cases}
$$
  For the initial condition $(t_0,x_0)=(0,c)$ with $c&gt;0$, we can define the solution $\varphi(t)=c-t$ for $t\le c$. Show that there does not exist a piecewise continuous solution, however, for $t&gt;c$.


Any help/proof would be appreciated!
Assume $\varphi$ exists for $t &gt; c$. Put $\psi := - \varphi$, then $\psi$ is a solution of $x' = \text{sgn}$. Now consider the sum of $\varphi$ and $\psi$ and their derivative in $t = c$...
Assume $\varphi$ exists for $t &gt; c$. Put $\psi := - \varphi$, then $\psi$ is a solution of $x' = \text{sgn}$. Now consider the sum of $\varphi$ and $\psi$ and their derivative in $t = c$...