The problem with both characterizations is that they ignore the underlying probabilities.
Recall that a random variable $X$ is a function that assigns real numbers to elements of the sample space. If a considerable part of the domain of $X$ has no probability, then the range of $X$ may have virtually any property whatsoever but that won't tell you a thing about the distribution of $X.$
Here are the mathematical details.
By definition, a random variable $X$ has a distribution function defined by $$F_X(x)=\Pr(X\le x)$$ for all numbers $x.$ $X$ is continuous if and only if $F_X$ is a continuous function everywhere.
As a counterexample to both (a) and (b), let $\Omega=[0,1]$ be the sample space of all real numbers between $0$ and $1$ inclusive with its usual Borel sigma-algebra. $\Omega$ is uncountable. Let $\mathbb P$ be the normalized counting measure on $\{0,1\}.$ This means the value of $\mathbb P$ on any event $\mathcal E\subset \Omega$ is the sum of two values: $0$ if $0\notin \mathcal E$ or $1/2$ if $0\in\mathcal E;$ plus $0$ if $1\notin \mathcal E$ or $1/2$ if $1\in\mathcal E.$ This is a standard way to model the flip of a fair coin, for instance.
Define a random variable by $$X:\Omega\to\mathbb{R},\quad X(\omega)=\omega.$$ By one standard definition, the range of $X$ is the smallest interval $[a,b]\subset\mathbb R$ for which $\mathbb{P}(X\in[a,b])=1.$ Clearly $0\in[a,b],$ $1\in[a,b],$ and $\mathbb{P}([0,1])=1,$ whence the range of $X$ is $[0,1].$
(Notice how this models the intuition in the introductory paragraphs: although $X$ takes on uncountably many possible values, the only values that have any nonzero probability are limited to just the finite set $\{0,1\}.$)
Although the range of $X$ is the uncountable set $[0,1],$ the distribution function $F_X$ is piecewise constant, jumping from $0$ to $1/2$ at $x=0$ and from $1/2$ to $1$ at $x=1.$ (This is the Bernoulli$(1/2)$ CDF.) $F_X$ is obviously not continuous at either point, even though (a) the range of $X$ is uncountable and (b) the sample space $\Omega$ is uncountable.