Let $X: \Omega \rightarrow \mathbb{R}^k$ be a random variable and let $F_X(x)$ be it's cumulative distribution function. In $\mathbb{R}^1$, $\mathbb{P}(X = x) = 0$ implies that $F_X(x)$ is continuous. Show that for $k > 1$, $F_X$ can have a discontinuity despite $\mathbb{P}(X=x) = 0$.
Here is my approach:
if we take $k = 2$ and let $X = (X_1, 0)$ where $X_1$ is a RV with CDF $F_{X_1}$ which is continous everywhere in $\mathbb{R}$. Then $X \in \mathbb{R} \times {0} = \{ x=(x_1, 0): x_1 \in \mathbb{R} \} \subset \mathbb{R}^2$.
From here on, what is left is to show that $F_X$ has uncountably many discontinouity pts while $\mathbb{P}(X = x) = 0$. However I am stuck here.