[25] Let \((t_0,x_0)\in \Omega \) be a point in an open set \(\Omega \subset \mathbb {R}\times \mathbb {R}^d\) and \(f:\Omega \to \mathbb {R}^d\) be a continuously differentiable function. Show that for some \(\epsilon > 0\) the initial value problem
\[ \dot {u}=f(t,u) \]
\[ u(t_0)=x_0 \]
has a unique continuously differentiable solution \(u(t):[0,\epsilon ]\to \mathbb {R}^d\) such that \((t,u(t))\in \Omega \) for all \(0 \le t \le \epsilon \). [Don’t just quote a result. State the theorem and give as complete and detailed a proof as you can.]