Flow-like functions associated to the solutions of the IVP $x'=f(t,x)\quad x(0)=x_0$

Question

I'm trying to prove the next result on flow-like functions for differential equations:

Let $f$ be a function and suppose that for every $u=(0,x_0)\in \mathbb{R}^+\times\mathbb{R}^n$ there is a maximal solution for the IVP $$x'=f(t,x)\quad x(0)=x_0$$ containing $u$, say $\phi(t,x_0)$. Consider the family of functions \begin{eqnarray} K_t:\mathbb{R}^n&\to&\mathbb{R}^n\\ x&\mapsto& K_t(x):=\phi(t,x) \end{eqnarray} with $t\in\mathbb{R}^+$. Then these functions satisfy the following conditions:

1. $K_0=0$
2. $K_{s+t}=K_s\circ K_t$
3. The map \begin{eqnarray} K:\mathbb{R}^+\times\mathbb{R}^n&\to&\mathbb{R}^n\\ (t,x)&\mapsto& K_t(x) \end{eqnarray} is continuous.

Here is my attempt. For the part two I showed that both $\phi(t+s,x_0)$ and $\phi(s,\phi(t,x_0))$ are solutions of the differential equation, so the conclusion follows from the uniqueness of the maximal solutions.

Any hints for the other two items? Specially the first one doesn't make any sense to me.