A function on $\mathbb{R}^d$ is Locally Lipschitz if when restricted to a compact subset it is Lipshitz see : https://en.wikipedia.org/wiki/Lipschitz_continuity .
I have been told that if a function $f:\mathbb{R}^d\to \mathbb{R}^d$ satisfies the following growth conditions :
\begin{align} \| f(x) - f(y) \| \leq C \| x-y \| \big( 1 + \| x \|^{r-1} + \| y \|^{r-1} \big), ~~~~~~~(1)& \\ \| f(x) \| \leq C \big( 1 + \| x \|^r\big).~~~~~~~(2)&~~~r= 1,2,3\ldots \end{align}
then it is Locally Lipshitz, $\textbf{Question :}$ does anyone know if this is true?
$\textbf{Question :}$ Equation $(2)$ says that the function $f$ is growing at most polynomially, what does $(2)$ tells us about $f$? Some intuition about the role that these conditions play when $f$ governs an ODE $\frac{d}{dt}x(t)=f(x(t))$ (or SDE) would also be greatly appreciated.