doublets is greater than $0$. To see that $P(Y=x)>0$ is an impossibility for any $x$, note that $Y$ being continuous means that $F(x)=P(Y\leq x)$ is continuous in $x$. Thus, since $P(a<Y\leq b)=F(b)-F(a)$, $$P(Y=x)=\lim_{\epsilon\rightarrow 0} P(x-\epsilon<Y\leq x+\epsilon)=\lim_{\epsilon\rightarrow 0}\Big( F(x+\epsilon)-F(x-\epsilon)\Big)=0.$$ Thus $P(X_i=X_j)=P(Y=0)=0.$ This works in the same way as does length. The length