thanks for reading.
Consider a one-dimensional dynamical system $\dot{x} = f(t,x)$. Let's call $\phi(t,t_0,x_0)$ the solution passing through $x_0$ at time $t_0$ (where $t$ is the time argument of such solution). For simplicity let us assume $t_0 = 0$. Poincare's map in one dimension can be simply defined as $P: x_0 \mapsto \phi(1,0,x_0)$. In other words, we map the initial value $x_0$ to the value (at time $t=1$) of the solution passing through $x_0$ at time $t_0 = 0$. Now, if uniqueness of solutions holds, then solutions can't cross and it's easy to see that Poincar\u00c3\u00a9's map is monotone and increasing.
Ok, the problem is: a friend of mine is studying such things for the bachelor degree and her teacher says that you also need 'continous and differentiable dependence of the solution from the initial value' in order to prove Poincare's map's monotonicity.
I can understand that such dependence can be used to prove the fact that the map $P$ is differentiable, but i can't see why this should help proving the monotonicity of Poincar\u00c3\u00a9's. Am i missing something?