dependence on initial conditions of solutions of ordinary differential equations

Let $E\subset W$ where $W$ is a normed vector space, $f\in C^1(E)$ is a continuous differentiable map $f:E\to W$. Furthermore consider the ordinary differential equation

$$\dot{x}=f(x)$$

with the initial condition

$x(0)=x_0$.

Let $x(t)$ be the solution of the above initial value problem defined as

$$x:I\to E$$

where $I=[-a,a]$. Then there exist $\delta >0$ such that for all $y_0\in N_{\delta }(x_0)$($y_{\mathrm{0}}$ in the $\delta$ neighborhood of $x_{\mathrm{0}}$) has a unique solution $y(t)$ to the initial value problem above except for the initial value changed to $x(0)=y_0$. In addition $y(t)$ is twice continouously differentialble function of $t$ over the interval $I$.