Is it true that $ Pr(X\leq a, Y\leq b)=1 \Leftrightarrow a=\infty, b=\infty $ for continuous random variables? Consider two random variables $X,Y$ both continuously distributed on $\mathbb{R}$. Also the random vector $(X,Y)$ is continuously distributed on $\mathbb{R}^2$.
It follow that
$$
Pr(X\leq a)=1 \Leftrightarrow a=\infty
$$
and
$$
Pr(Y\leq b)=1 \Leftrightarrow b=\infty
$$
Without imposing further assumptions on the joint distribution of $X,Y$, can we also say that 
$$
Pr(X\leq a, Y\leq b)=1 \Leftrightarrow a=\infty, b=\infty
$$
?

Edit following suggestion in answer below: I should replace the starting sentence "Consider two random variables $X,Y$ both continuously distributed on $\mathbb{R}$. Also the random vector $(X,Y)$ is continuously distributed on $\mathbb{R}^2$." 
with
"Consider two random variables $X,Y$ both continuous and with non-negative density on $\mathbb{R}$. Also the random vector $(X,Y)$ is continuous with non-negative density on $\mathbb{R}^2$."
 A: No. Let $X$ and $Y$ be uniformly distributed on $[0,1].$ Then $P(X\le 2,Y\le 2) =1).$
Your assertion that it is true for single random variables is similarly wrong. Unless you are using a different definition of "continuously distributed on $\mathbb R$" than what is standard.
edit
In response to the clarification, yes, if $(X,Y)$ has positive density my on all of $\mathbb R^2$ then we have $P(X\le x,Y\le y) < 1$ if $x$ or $y$ are less than $\infty.$  If $y < \infty$ then we can integrate positive density over the region $X\le x, Y>y$ and get a positive probability outside the original region.
But having positive density on $\mathbb R^2$ does not follow from each variable having positive density on $\mathbb R.$ For instance we can have $X$ a standard normal and $Y=-X.$ Then both are marginally standard normal but two quadrants of $\mathbb R^2$ have zero density.
But your statement still holds in this weaker situation since if there is zero probability of that $X>x$ or $Y>y,$ then there must be zero probability that $X>x$ and zero probability that $Y>y.$
