I'm studying the Probability from the book Probability, Alan F. Karr.
Proposition 2.29.
If X = (X1 , .. . , Xd) is absolutely continuous, then for each i, Xi is absolutely continuous, and $f_{X_i}(x)=\int_{-\infty}^{\infty}...\int_{-\infty}^{\infty}f_X(y_1,...,y_{i-1},x,y_{i+1},...,y_d)dy_1...dy_{i-1}dy_{i+1}...dy_d$
(...)
The converse to Proposition 2.29 is not true: if X1 is uniformly distributed on [0, 1] and X2 $\equiv$ X1 , then (X1, X2) is not absolutely continuous even though the components are. The converse is true, however, when the components are independent, as shown in Corollary 3.6.
I couldn't understand why the example X2 ≡ X1, X1 is uniformly distributed proves that the converse isn't true. I think my problem is in the X2 ≡ X1, is (X1,X2) $\equiv$ (X1,X1)?
If so, when I represent $F_{(X_1,X_2)}$ using three dimensions it whould be a line segment, from $(0,0,0)$ to $(1,1,1)$, right?