Multivariable definition of continuity in Buck Advanced Calc. I was working through Buck's Advanced Calculus and found his definition of continuity strange compared to the definition in Rudin's PMA.
I thought his definition was useful, but is it wrong when compared to the definition in baby Rudin? I realize Rudin is working in metric spaces, and was hoping to quickly go through Buck and do baby Rudin. I've read, "you should have the definitions down cold."
Below is a copy of Buck's definition. It isn't the usual $\varepsilon$ - $\delta$ definition, per se.
D is a region in the plane.

Definition 1: A numerical-valued function $f$, deﬁned on a set $D$, is said to be continuous at a point $p_o \in D$ if, given any number $\varepsilon > 0$, there is a neighborhood $U$ about $p_o$ such that $\vert f (p)- f (p_o)\vert\lt \varepsilon$ for every point $p\in U \cap D $. The function $f$ is said to be continuous on $D$ if it is continuous at each point of $D$.

 A: They are synonym. Here's 
Rudin's definition quoted, and compared to Buck's definition

Suppose $X, Y$ are metric spaces, $E \subset X$, $p \in E$ and $f$ maps $E$ into $Y$...

Take $X = \mathbb{R}^2, Y = \mathbb{R}$ with usual Euclidean metric $d_X(\textbf{x},\textbf{y}) = \sqrt{(x_1-y_1)^2 + (x_2 - y_2)^2}$, $d_Y(x,y) = |x-y|$, and denote $E$ by $D$, and so $f: D \to \mathbb{R}$

... Then $f$ is said to be continuous at $p$ if for every $\epsilon > 0$ there exists a $\delta > 0$ such that $d_Y(f(x), f(p))>\epsilon$ for all points $x \in E$ for which $d_X(x,p) < \delta$

Said another way, $f$ is continuous at $p_0 = p \in D$ if, given $\epsilon > 0$, there is a neighborhood $U$ about $p_0$ (i.e. a set of points $x$ such that $d_X(x,p) < \delta$ for some $\delta > 0$, see page 32 for def. in Rudin) satisfying $d_Y(f(x), f(p_0)) = |f(x) - f(p_0)| < \epsilon$ for every points  $x \in D$ (where $f$ is defined), but for which $d_X(x, p_0) < \delta$ (i.e. $x \in U$ the neighborhood about $p_0$) hence the intersection $x \in U \cap D$ in Buck's definition.
