I am trying to figure out where the cdf has a jump. I believe there is a jump at $x=\frac{1}{2}$, but it is obviously wrong and I am not sure why that is the case. I would appreciate if someone could provide an explanation on how it is done for any random variable and how the sketch looks like.
$Y(\omega)=\omega$ for $\omega\in[0,\frac{1}{2}]$ and $Y(\omega)=\frac{1}{4}$ for $\omega\in(\frac{1}{2},1]$
For $0 \le x \le 1/2$, $\mathbb P(\omega \le 1/2, X \le x) = \mathbb P(\omega \le x) = x$. Then there is an additional probability of $1/2$ that $X = 1/4$ with $\omega > 1/2$. So the cdf is $$ F(x) = \mathbb P(Y \le x) = \cases{0 & if $x < 0$\cr x & if $0 \le x < 1/4$\cr 1/2 + x & if $1/4 \le x \le 1/2$\cr 1 & if $x > 1/2$\cr}$$