A question about right continuous functions on $\mathbb{R}^d$.

Question

I am reading Rick Durrett's Probability Theory and Examples. In Chapter 1 of the book (page 6), there are some contents about right continuous on $\mathbb{R}^d$. It says

(ii) $F$ is right continous, i.e., $\lim_{y \downarrow x} F(y) = F(x)$ (here $y \downarrow x$ means each $y_i \downarrow x_i$).

To me, this is a little unclear that $\lim_{y \downarrow x}F(y) = F(x)$ in terms of the $\epsilon-\delta$ argument. Can I define this as following?

"For every $\epsilon > 0$, there exist constants $m_i$'s such that if $x_i \le y_i < m_i $, then $\vert F(y_1, y_2, \cdots, y_d) - F(x_1, x_2, \cdots, x_d) \vert < \epsilon$.", where $x = (x_1, x_2, \cdots, x_d)$ and $y = (y_1, y_2, \cdots y_d)$.

Answer 1

Sure. Alternatively, the restriction of $F$ to $\prod_{i=1}^d [x_i,\infty)$ is continuous at $x=(x_i)_{1≤i≤d}$ for every $x$.