How can two random variables are continuous infers that their jointly random variable is continuous [closed]

Question

We assume that $\forall a,b$ suchthat $a^2+b^2>0$, $aX+bY$ is continuous random variable.

But we don't assume that $X$ and $Y$ are independent.

My question is the following:

Is it true that the jointly random variable $(X,Y)$ is continuous?

If true,how to show ?if not,please give a counter-example or under which conditions it will be true.

Thanks.

Answer 1

Counterexample: let $X = \cos(\Theta)$, $Y = \sin(\Theta)$, $\Theta$ uniform on $[0,2\pi]$.