This is a true or false question.
Let $f(x,t)$ continuous and $\dot{x}=f(t,x)$ an ODE with unicity of solution. So, given $\epsilon >0$, exists $\delta >0$ such that
$$|g(x,t)-f(x,t)|<\delta\; \forall (x,t)\in\mathbb{R}^{n}\times\mathbb{R}\implies |\gamma(t)-\eta(t)|<\epsilon, $$ where $\gamma$ and $\eta$ are the solutions of respectives ODEs with $\gamma(0)=\eta(0)=0$.
I think that taking $f(t,x)=0$ and $g(x,t)=x+1$ I can give a counterexample for $n=1$, because $\gamma(t)=0$ and $\eta(t)=e^{t}-1$ are not functions "close" to each other. Is my attempt right? How can I write that?