Are those Locally Lipschitz definitions equivalent? Let $f:\mathbb{R}^{+}\times \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ be locally Lipschitz in the sence that 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 $$|f(t,x)-f(t,y)|\leq \ell (t,|x|+|y|)|x-y|$$
(another similar definition is given by Sontag in Mathematical Control Theory)
Is this equivalent to the classic definition?
f is locally Lipschitz; satisfies the lipschitz condition
$$|f(t,x)-f(t,y)|\leq L|x-y|$$
for all $(t,x)$ and $(t,y)$ in some neighborhood of $(t_0,x_0)$, $L\geq 0$.
My thoughts:
i've tried restricting in a compact set A which can be covered by a finite number of neighborhoods
$$A\subset N(a_1,r_1)\cup\cdots\cup N(a_k,r_k)$$
Then i can get for each $i=1,...k$ the lipschitz constants $L_i$ since
$$|f(t,x)-f(t,y)|\leq L_i |x-y|, \forall (x,y)\in N(a_i,r_i)$$
Is there a way to patch all these constants and find a dominating continuous function over them?
 A: It seems that the variable $t$ plays no rôle here; so we may as well omit it.
The essential difference between the two definitions is the following: The first definition is only applicable to functions $f$ defined on all of ${\Bbb R}^n$, whereas the "official" definition can as well be applied to functions defined only in some open domain $\Omega\subset{\Bbb R}^n$. An example: The function $f(x):=|x|^{-2}$ is defined in $\Omega:={\Bbb R}^n\setminus\{0\}$ and locally Lipschitz in $\Omega$, but it is not locally Lipschitz according to the first definition.
For functions $f:\>{\Bbb R}^n\to{\Bbb R}^n$ the two definitions are equivalent:
Assume that $|f(x)-f(y)|\leq\ell\bigl(|x|+|y|\bigr)|x-y|$, and let a point $x_0\in{\Bbb R}^n$ be given. Then
$$|f(x)-f(y)|\leq\ell(2|x_0|+2)|x-y|$$
for all $x$, $y\in B_1(x_0)$.
Conversely: Assume that $f$ is locally Lipschitz according to the "official" definition, whence continuous. Let an $r>0$ be given. I claim that there is a finite $\ell>0$ with $|f(x)-f(y)|\leq\ell|x-y|$ for all $x$, $y\in B_r$. If not, one could find two sequences $(x_n)_{n\geq1}$, $\>(y_n)_{n\geq1}$ in $B_r$ with $$|f(x_n)-f(y_n)\geq n|x-y|\qquad(n\geq1)\ .\tag{1}$$ After sieving we may assume $x_n\to\xi\in B_r$, $\>y_n\to\eta\in B_r$, and $(1)$ then implies $\xi=\eta$. In this case $(1)$ contradicts the assumption that $f$ is Lipschitz in some neighborhood of $\xi$. 
Denote by $\ell(r)$ the infimum of all admissible $\ell$s for $B_r$. It is then easy to see that $\ell(r)$ makes $f$ locally Lipschitz according to your first definition.
