Consider the system $\dot{x}=f(t,x)$, $x\in \mathbb{R}^{n}$, $t\geq 0$ with $f:\mathbb{R}^{+}\times \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$, $f(t,0)=0$ and $f(t,x)$ is locally Lipschitz. Then there exists a positive $C^{0}$ function $\ell :\mathbb{R}^{+}\times \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that for each fixed $s\geq 0$ the mappings $\ell(\cdot,s)$ and $\ell(s,\cdot)$ are nondecreasing and the following holds $$|f(t,x)-f(t,y)|\leq \ell (t,|x|+|y|)|x-y|$$
Any help or tips would be appriciated. thanks in advance