I have a Lipschitz function $X=X(t)$ with the property that, at all points $t$, the right derivative $\lim_{\epsilon \downarrow 0} \epsilon^{-1}(X(t+\epsilon)-X(t))$ exists and is given by $f(X)$ for some (discontinuous) function $f$. (Of course, at all regulat points, i.e. almost everywhere, this is then the ordinary derivative.) Is it true that such $X$ is necessarily unique?

Of course, if I just specify that a Lipschitz function satisfies $X'(t) = f(X(t))$ at all regular points, the solution doesn't have to be unique; but I'm requiring that this hold for the right derivative at *all* times, which intuitively seems like it ought to work.