Discrete and Continuous variables. What is the definition? The definition of a continuous variable in our class seems to be, well,  not a definition,  as there are exceptions not included in its definition.
I am a 4th year math student and find it appalling that such a rinky dink thing can be a definition.
Could someone possible give me a definition that is capable of differentiating between continuous variables and discrete ones that is completely accurate or give me a list of all the continuous variables that need to be treated as discrete and when. So far I have that money is somehow discrete as well as time sometimes?
Concentrations I'm not sure about (e.g parts per million) but weight, length and temperature seem to be continuous. Time seems to be completely confused.
They claim that a continuous random variable is one that would take an infinite amount of time to list all possible values, whereas a discrete one is one that can be counted.

Here is the most elementary definition I can think of.
If $X$ is a random variable then it has a "distribution function", also known as as CDF. If we let $F(t)$ denote the distribution function then by definition $F(t) = P(X\leq t)$. If this distribution function is continuous (as a function in ordinary calculus) then we say that $X$ is a continuous random variable.
We say that $X$ is a discrete random variable if the "support of $X$" consists of the integers. Recall the "support of a random variable" is the set of possible values the random variable may take. Let us denote the support by $\text{supp}(X)$. Therefore, we say that $X$ is a "discrete random variable" if $\text{supp}(X)\subseteq \mathbb{Z}$. Even more simply, we say $X$ is discrete if the value of $X$ is always some integer value.
Note, the above definition for "discrete" is not the most general definition, but as an initial definition it is good enough. The most important discrete random variables are integer valued anyway. Hence you do not need to be concerned about (yet) what it means to be a "countable set" at the moment.
If $X$ is a discrete random variable then its CDF will look like a staircase, the CDF will be piecewise constant.
