5.1.6 problem 7

a. F(x) is continuous at its given endpoints: $F\left(1\right)=\frac{2}{\pi }\arcsin \left(1\right)=\frac{2}{\pi }\frac{\pi }{2}=1$ and $F\left(0\right)=0$, and F(x) is differentiable between 0 and 1:

b. $F^{\text{'}}\left(x\right)=f\left(x\right)=\frac{2}{\pi }\frac{d}{\mathrm{dx}}\left(\arcsin \right(\sqrt{x}\left)\right)=\frac{2}{\pi }\left(\frac{1}{\sqrt{1-x}}\right)\left(\frac{1}{\sqrt{x}}\right)$

This is a valid PDF despite the discontinuities at 0 and 1 as the integral of $f\left(x\right)$ from 0 to 1 converges, and $f\left(x\right)$ is always positive - the same reason why $\frac{1}{\sqrt{x}}$ has a discontinuity at $x=0$ but can be integrated from 0 to any positive real number.