Let $\epsilon >0$ and $f\in \mathcal{C}^1(\mathbb{R}^{d+1})$, and we consider the following Backward Foward ODE : $$\begin{cases} \dot z^{\epsilon}(t)=f(z^{\epsilon}(t),t), & t\in [0,2]\\ z^{\epsilon}({\epsilon})=z_0\in \mathbb{R}^d .\end{cases}$$ According to Picard's existence theorem, there is a local solution of this problem $(z^{\epsilon},J^{\epsilon})$ defined on an open interval $J^{\epsilon}$.
Now, my question is that if we assume that there is $c\in[0,2]$ for all $\epsilon >0$, such that $c\in J^{\epsilon}$ : can we prove that $z^{\epsilon}$ is bounded on the interval $[\epsilon,c]$ ?