Chapter 6, Problem 6.12TE

Show that the jointly continuous (discrete) random variables $X_1,\mathrm{...},X_n$ are independent if and only if their joint probability density (mass) function $f\left(x_1,\mathrm{...},x_n\right)$ can be written as $f\left(x_1,\mathrm{...},x_n\right)={\displaystyle \underset{i=1}{\overset{n}{\prod }}{g}_i\left(x_i\right)}$ for nonnegative functions $g_i\left(x\right),i=1,\mathrm{...},n$.