Let $X$ be a random variable. I suspect that if the CDF $F_X$ is continuous, then $\text{im}(X)$ is uncountable. I reason as follows:
Attempt: as $\lim_{x\to-\infty}F_X(x) = 0$ and $\lim_{x\to+\infty}F_X(x) = 1$, we may choose $a<b$ both in the image of $F_X$. By continuity and the Intermediate Value Theorem, the function $F_X$ attains all the uncountably many values in between $a$ and $b$. Thus, the expression $$F_X(c) = P(X\le c)$$ also attains uncountably many values as $c$ varies in $\mathbb{R}$ and here I would like to say the former implies that $X$ also attains uncountably many values, but I'm struggling to make this last bit of the argument rigorous.