How can I properly define $R$ and $R^2$?

Question

I am looking for a definition of $R$ and $R^2$ to use in a report I'm writing.

I have the following line where I am struggling to figure out what $R$ and consequently $R^2$ means:

$f$ and $f_x$ are continuous in the rectangle: $R = \{(x,t):a\leq x \leq b, c \leq t \leq d\}$

Any help would be greatly appreciated. I'm guessing $R$ is a region of 2 dimension as a starting point?

Answer 1

$R$ is a rectangle in the plane $\mathbb{R}^2$. Its corners are $(a,c)$ and $(b,d)$. Note that for a point $(x,y)$ to be in a rectangle, it must be between any two opposite corners, which is the case if and only if both coordinates (i.e., both $x$ and $y$) lie between the respective coordinates of the corners. This is exactly how $R$ is defined.

Answer 2

Nicolas, gave you the answer, if you wanted to talk about the Cartesian product of the reals with the reals (usually called the Cartesian plane in this set analysis) for $R^2$. Usually we use the notation \mathbb{R} which produces the double struck $\mathbb{R}$ for the reals. In this case, I believe the definition of $R^2$ is the Cartesian product of the the set R with itself. R appears to be a bounded shape, called a rectangle. The set notation definition for $R^2$ could be as follows:$$R^2 = \{(x_1,t_1,x_2,t_2):a\leq x_1,x_2\leq b ; c\leq t_1,t_2 \leq d\}$$

This new shape, is called a box (it is only a cube if d-c=b-a)