What is the value of multivariate CDF when one of the arguments is at the lower end of the support? e.g. $F(a,0)$

Question

Suppose there are two random variables $X$ and $Y$ with continuous support $[0,1]$ and bivariate distribution of $X$ and $Y$, $F$.

What would be the value of CDF $F(x,0)$ when $x\in[0,1]$? Is it just 0 because there is no atom in a continuous distribution?

Answer 1

Yes. $P\{X,\leq x,Y\leq 0\}\leq P\{Y\leq 0\}=0$.