 Solution

Let $W=U^2$. On $0\le U\le 1$, squaring preserves order. Therefore for $0\le w\le 1$,Including the tails gives the cumulative distribution functionDifferentiating on the interior of the support gives the probability density functionThe divergence near zero is integrable: ${\int }_0^1(2\sqrt{w}{)}^{-1}dw=1$. There is no point mass at zero, and the assigned value of a probability density function at a single endpoint does not affect the probability distribution.