I recently learned that whenever $f:\Bbb R^n\to\Bbb R^n$ and $f\in C^1$ then the given the system defined
$$ \begin{array}{rcl} \dot{x} & = & f(x) \\ x(0) & = & x_0 \end{array} $$
the solutions can be viewed as a group action on $\Bbb R^n$ (a flow). In particular, I was interested by the "reversibility" condition, ie. that for every map $\phi_t$ there is an inverse map $\phi_{-t}$ such that $\phi_{-t}\phi_t$ is the identity map.
I want to verify my example of a function $f\notin C^1$ and corresponding solutions which are not reversible. My idea is "a ball rolling off a cliff". Let $f:\Bbb R\to\Bbb R$ defined by
$$ f(x) = \begin{cases} 1 & \text{if } x < 0 \\ 0 & \text{if } x \geq 0 \end{cases} $$
Let the initial condition $x(0)=x_0<0$. I believe that one solution to this system (we cannot guarantee uniqueness) is
$$ x(t) = \begin{cases} x_0 + t & 0 \leq t < |x_0| \\ 0 & t \geq |x_0| \end{cases}$$
So it just rolls towards the origin then stops. Now consider this problem in reverse: we have an initial point $x(0)=0$ and we have $f(x(0))=0$, so the solution is $x(t)=0$ for all $t$. Clearly not the reverse of our first solution, i.e. "the ball cannot roll back up the cliff". So, is this a proper example?