The definition of locally Lipschitz

Marsden's Elementary Classical Analysis seems to indicate this definition:

• A function $f:A{\subset}\mathbb R^n\to\mathbb R^m$ is locally Lipschitz if

  for each $x_0{\in}A,$ there exist constants $M{>}0$ and $\delta_0{>}0$ such that $$||x-x_0||<\delta_0\implies||f(x)-f(x_0)||\leq M||x-x_0||.$$

Here's a scan of the first edition of the text, where the only change from the second (latest) edition referred to above is that the last sentence now reads "This is called the local Lipschitz property" (emphasis mine).

Two questions:

1. Is the correct inequality $M\geq 0$ or $M>0$?

2. Does $M$ depend on $x_0$, like $\delta_0$ seems to?

EDIT: After pondering further, I've revised the definition to this:

A function $f:A{\subset}\mathbb R^n\to\mathbb R^m$ is locally Lipschitz at $x_0{\in}A$ if

there exist constants $\delta{>}0$ and $M{\in}\mathbb R$ such that for each $x{\in}A,$$$||x-x_0||<\delta\implies||f(x)-f(x_0)||\leq M||x-x_0||.$$

Unlike regular/global Lipschitz, local Lipschitz can be defined at a point, and implies pointwise continuity.