Given iid continuous random variables$\displaystyle X_1, X_2,...,X_n$, let $\displaystyle Y=g(X_1,...,X_n)$ be some well-behaved function (e.g., sum) of these r.v.'s and $\displaystyle f$ be the joint pdf of $\displaystyle X_1, ...,X_n,Y$. Do we have $\displaystyle f(x_1,...,x_n,y)=f(x_1,...,x_n)$ if $\displaystyle y=g(x_1,...,x_n)$ and $\displaystyle f(x_1,...,x_n,y)=0$ if $\displaystyle y\neq(x_1,...,x_n)$? If so, how to prove? This statement is obviously true for pmf, but I have no idea for pdf.

Thanks a lot!