A mapping or solution $g(x_0,t_f):x_0 {\rightarrow} x$ of an ode $\dot{y}=f(y,t)$ is a Homeomorphism for any constant $t_f$. I have a mapping which is also a solution for an ode, but with continues, non-constant $t_f$, is this mapping still a Homeomorphism?
For example given the mapping $g(x_0,t_f)=x_0e^{t_f}$. If $x_0=(1,2)$ and $t_f=\frac{x_0}{2} $ ($t_f$ is a continues function of $x_0$). Than $g(x_0,t_f):(1,2){\rightarrow} (e^\frac{1}{2},2e)$ is a maping of an open-set to an open-set i.e. a Homeomorphism.
How can I test/prove if such a mapping is a Homeomorphism when the analytic mapping is not available (because there is no explicit solution for the ode)?