= Solution
Let $W=U^2$. On $0\leq U\leq1$, squaring preserves order. Therefore for $0\leq w\leq1$,
$$
F_W(w)=\mathbb P(U^2\leq w)=\mathbb P(U\leq\sqrt w)=\sqrt w.
$$
Including the tails gives the <cumulative distribution function>
$$
\boxed{F_W(w)=\begin{cases}0&w<0,\\\sqrt w&0\leq w\leq1,\\1&w>1.\end{cases}}
$$
Differentiating on the interior of the support gives the <probability density function>
$$
\boxed{f_W(w)=\begin{cases}\dfrac1{2\sqrt w}&0<w<1,\\0&\text{otherwise}.\end{cases}}
$$
The 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>.
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