Check for independence of variables when the density (or distribution) is known. This question is closely related to a previous one:
Determine correlation and independence when only the joint density is given?
Nonetheless, the setting is reproduced below. 
The joint pdf of $X = (X_1,\ldots,X_n)$ is:
$$f_{X}(x_1,\ldots,x_n)=\begin{cases} Ar^2,&0 \le r \le R\\[0.2cm]  0,& \text{ otherwise }\end{cases}$$
where $r = \sqrt{x_1^2 + \ldots + x_n^2}$ and $A,R$ are constants.
I would like to know if the $X_j$'s are independent. Since the density is a sum over the $X_j$'s and the sum is bounded above by $R$ (and below by 0) , I would think the $X_j$'s are dependent. However, I have three concerns regarding this intuition.


*

*How do I formally argue for (in)dependence by examining the density, and the distribution? 

*If I cannot formally argue by looking at the density/distribution alone, how do I check for independence of continuous variables? It does not seem feasible to check for the condition
$$P(X_1) = P(X_1 |X_2,\ldots,X_n)$$

*Can I still claim the $X_j$'s are independent if they are a sum, but the sum is not bounded above/below by some value?
Please address each one in turn!
 A: It seems -looking at point 1 and the comments- to me that you didn't quite understand the answer (and comments) to the previous question, even informally. 
Quick example: we are told that two variables $(X,Y)$ have some joint density $f_{X,Y}$ with support (that is, taking strictly positive values) over a unit circle centered in the origin: $f(x,y) > 0 \iff x^2+y^2<1$. This immediately says that the variables are NOT independent. 
Why? Because the marginals $f_X$ $f_Y$ will have support on the intervals $I_x = [-1,1]$ and $I_y=[-1,1]$, and then, for any $(x,y) \in I_x \times I_y$ (that, is for any point inside the rectangle -  whichs is cartesian product of the two intervals) we should have, $f_{X,Y}(x,y)=f_X(x) f_Y(y) >0$,  and zero otherwise . That is: the support should be a rectangle, and with sides parallel to the axes. 
This is just an example, the marginals need not be intervals. But, in general, what remains true is that the support of the joint density (the region where it takes positive values) must be the product (the cartesian product!) of the supports of each variable. If not a rectangle (perhaps unbounded) at least a set of rectangles. Never a circle or a circular sector, or other shapes.
Regarding the other points: the criteria above is a first necessary (not sufficient!) condition for independence, quite elementary but useful. It says nothing if the support is all the plane, for example.
Failing that, you need to apply the definition, and either check that the joint distribution factorizes as the product of the marginals, or compute the conditional. 
Nothing much can be said in general. (And I wonder why you emphasize that the variables are continuous , that seems pretty irrelevant).
The point 3 makes no sense to me. What does "they are a sum" mean? The sum of several variables is a single variable, so I have no idea what are the variables we are checking for independence then.
