Let $g \in \mathcal{C}(\mathbb{R}^{n},\mathbb{R}^n)$, and $c \in \mathbb{R}^+$, such that $0<a<a+\epsilon \leq T$ (where $\epsilon >0$).
and we define the following ODEs : Assuming the existence of an absolutely continuous solution $x(.)$ defined on $[0,T]$. $$\begin{cases} \dot x(t)=g(x(t)) & t\in[0,T]\\ x(a)=x_a, \end{cases} $$ Now, we introduce a small perturbation, and I want to prove that $x_{\epsilon}$ is defined at $a$ for small $\epsilon$ $$\begin{cases} \dot x_{\epsilon}(t)=f(t,x_{\epsilon}(t)) & t\in[0,b]\\ x_{\epsilon}(a+\epsilon)=(1+\epsilon)x_a, \end{cases} $$ We have the existence of a neighborhood of $(a+\epsilon)$ : $]a+\epsilon-\nu,a+\epsilon+\nu[$ for some $\nu>0$. But why is that this neighberhood can contain $a$ (since $a+\epsilon$ goes to $a$ when $\epsilon$ goes to $0$).