Defining continuous random variables via uncountable sets At several sources I have encountered the following two definitions of a continuous random variable associated with uncountable sets:
a) uncountable range: 
The random variable X is continuous if its range is uncountable infinite/set of possible values is uncountable infinite.
b) uncountable sample space: 
The random variable X is continuous if the sample space is uncountable infinite.
I have already learned that they are wrong but dont understand why. Hence, my questions would be how they relate to each other and in particular why they are wrong definitions?
 A: 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.
A: Well, even if the range (or support set) of the random variable $X$ is uncountable, $X$ do not necessarily have a density. The answer by @Sebastian mentions measure, and specifically counting measure. But counting measure on an uncountable set isn't very useful, for instance, it is not $\sigma$-finite. So not very useful in probability.
There is an interesting counterexample, the Cantor distribution have support on an uncountable set --- the Cantor (middle-third) set, but do not have a density, so is not absolutely continuous. Neither is it discrete, it is singular. See How to sample from Cantor distribution?,      Is probability theory the study of non-negative functions that integrate/sum to one?  and search ...
Such singular distributions are not common in statistics (except as counterexample), but are ubiquitous in other areas. See singular distributions applications and instances. A case in point is dynamics, with the famous Smale's horseshoe, where distributions supported on dynamical Cantor sets abound.
A: Consider the example where your sample space $\Omega$ = $\mathbb{R}$. This is uncountably infinite. However whether the RV is continuous depends on the used measure. If you would use $\mu = \#$ (i.e. the counting-measure) you could still easily defined a density with respect to $\mu$ that would induce a discrete distribution.
In general whether a distribution is discrete or continuous depends on the distribution function. It can of course also be a mixture of both (or to make things even weirded be 'singular', see e.g. the Cantor-distribution).
