Conditions to be a Joint Discrete Distribution Function I am reading a paper on modeling the dependencies in discrete distribution functions, and am having a hard time understanding the following.  Let us define: $$r \leq min(p,q)$$
$$ B(u,v) = \begin{cases}
0 & \text{if }u=0 \ \text{or } v=0 \\
r & \text{if }(u,v) \in (0,p] \times (0,q] \\
p & \text{if }(u,v) \in (p,1] \times (0,q] \\
q & \text{if }(u,v) \in (0,p] \times (q,1] \\
1 & \text{if }(u,v) \in (p,1] \times (q,1] \\
\end{cases} $$
$$ C(u,v) = \begin{cases}
r & \text{if }(u,v) \in [0,p) \times [0,q) \\
p & \text{if }(u,v) \in [p,1] \times [0,q) \\
q & \text{if }(u,v) \in [0,p) \times [q,1] \\
1 & \text{if }(u,v) \in [p,1] \times [q,1] \\
\end{cases} $$
The paper then says that $C(u,v)$ is a distribution function, while $B(u,v)$ is NOT a distribution function.  
Can anyone explain why this is the case? I do notice that C(u,v) is defined in a right continuous manner (which is typically how distribution functions are defined?), is that why?  Or is there something else that I am missing.  I can provide more context if necessary, but it seems to me that this is a standalone thing.
Thanks!
 A: If $B$ and $C$ are supposed to be joint cumulative probability distribution functions (joint CDFs), then you need to be aware that a joint CDF $F(\cdot,\cdot)$ must satisfy the "rectangle constraint" (a name that I just made up and so don't bother Googling for it). This rectangle constraint says that for all real numbers $a,b,c,d$ such that $a<c, b<d$, $F(c,d)-F(a,d)-F(c,b)+F(a,b)$, which equals
$P\{a < X \leq c, b < Y \leq d\}$ must be nonnegative:
$$F(c,d)-F(a,d)-F(c,b)+F(a,b) = P\{a < X \leq c, b < Y \leq d\} \geq 0.$$
For discrete random variables in particular, if $F(u,d)$ has constant 
value $p$ for $a \leq u <c$ and so does $F(c,v)$ have fixed value $q$
for all $b \leq v < d$, then $F(c,d)$ must be larger than $\max(p,q)$.
As an example, consider the function
$$B(u,v) = \begin{cases} 0, & u<0, v<0,\\
0, & 0 \leq u < 1, 0 \leq v < 1,\\
1, & \text{otherwise}
\end{cases}$$
which is right-continuous in both variables and has all the properties
that one might desire of a joint CDF except that
$$B(1,1)-B(0,1)-B(1,0) + B(0,0) = -1$$ and so is disbarred from
the ranks on joint CDFs by the rectangle constraint.
A: Probability distribution functions are defined so as to be right continuous ... unless you're in Russia, in which case they are defined so as to be left continuous.
Leo Breiman, best known to youngsters as the inventor of random forests and the person who coined the term "bagging", was perhaps a Commie, based on, for example, his 1973 book "STATISTICS: With a View Toward Applications", in which he uses the Russian, i.e., left continuous, convention, and never bothers informing the reader that he is departing from the standard convention outside of Russia.  Or look at Definition 2.21 at the bottom of p. 25 in his 1968 book "Probability".
