Copulas: Understanding U & V-P(U≤u)=C(u,1)=u?

[iomtt6076]

I'd appreciate it if someone could help clear up something I'm not understanding in a textbook I'm studying:

Given a copula C(u,v), we have $$P(U\leq u)=C(u,1)=u$$.

But why isn't it $$P(U\leq u)=C(u,\infty)=u$$? Isn't it true that $$C_U(u)=P(U\leq u)=\lim_{v\to\infty}C(u,v)$$?

 

[bpet]

But why isn't it $$P(U\leq u)=C(u,\infty)=u$$? Isn't it true that $$C_U(u)=P(U\leq u)=\lim_{v\to\infty}C(u,v)$$?

Yes it is, since copulas are also joint distribution functions.

 

[iomtt6076]

Okay, thanks; I guess the textbook should have explicitly said copulas were defined on [0,1]x[0,1].