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.

A random variable is a function that maps a sample space to the real numbers. A continuous random variable is such a function such that it can take on any value in an interval - not any arbitrary interval, but an interval which makes sense for any particular random variable under consideration. A discrete random variable is a random variable that can only assume a finite or countably infinity number of distinct values.
For reference, see Mathematical Statistics with Applications, 5th Ed., by Wackerly, Mendenhall, and Scheaffer. The random variable is defined as Definition 2.11 on p. 65. The discrete random variable is defined as Definition 3.1 on p. 76. The continuous random variable is defined on p. 136.
In the referenced textbook, I have never seen exceptions to these definitions, except perhaps the so-called "mixed" random variables that are partly discrete, partly continuous.
