We know that random variables that is of uniform distribution can only have 1 numerical value in a certain range and 0 otherwise.
But what happens if we multiply two uniformly distributed random variables? With two different ranges?
For example lets take $X$ ~ Uni($-1, 1$), $Y$ ~ Uni($0, 1$) and $U = XY$
From what I understood, we have: $$ f_{XY}(xy) = f_{U}(u) = \begin{cases} 1/2 & ; -1<X<1,& 0< Y<1 \\ 0 & ; \,\text{otherwise} \\ \end{cases} $$
How can this function have a pdf when it is known that basically $U$ is also a uniformly distributed random variable?
If $X$ and $Y$ are two independent, continuous random variables, described by probability density functions $f_{X}$ and $f_{Y}$ then the probability density function of $U=X Y$ is
$$ f_{U}(u)=\int_{-\infty}^{\infty} f_{X}(x) f_{Y}(u / x) \frac{1}{|x|}\mathrm d x . $$ Now $X\sim$Unif$(-1,1)$, hence $$f_X(x)=\begin{cases}1/2&x\in(-1,1)\\0&\text{else}\end{cases}$$ and since $Y\sim$Unif$(0,1)$, hence $$f_Y(y)=\begin{cases}1&y\in(0,1)\\0&\text{else.}\end{cases}$$ Substitution yields your pdf.