Does the Concept of "Individual Probabilities" Exist? This is a concept I have always struggled to understand:

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*When talking about Discrete Probability Functions, we usually have no problem assigning a probability to a single event. For example, when rolling a 6-sided die, we can say that the probability of encountering any of sides will have a certain probability.


*However, when talking about Continuous Probability Functions, we are told that that idea of "individual probabilities" does not exist. For example, consider a random variable "X" that has a Normal Distribution with Mean = 0 and Sigma = 1. We can find out the probability of this random variable "X" taking values between 2.9 and 3.1 - but we can not find out the probability of this random variable "X" taking an exact value of 3.
I always wondered - what is the formal reason behind this?
For example:

*

*We know that the "Mode" represents the most common number of a sample - and the "Mode" is a single number.


*We could then find out the probability of a random variable taking values between "mode + 0.0001" and "mode - 0.0001" (e.g. via an integral) - and try to see if this number converges for smaller and smaller deviations from the "mode", thus finding out the probability of encountering exactly the mode.
I am aware that this logic that I have described is most likely flawed - but could someone please help me understand why the concept of "individual probabilities" does not exist in the continuous sense?
Thanks!
 A: 
However, when talking about Continuous Probability Functions, we are told that that idea of "individual probabilities" does not exist.

I don't know who told you this but it's not true. As mentioned in the comments, the individual probabilities exist, they're just equal to zero and so not very informative. In particular they cannot be used to define a sensible mode.
What does make sense is that if $X$ is a random variable with a continuous probability density function $f(x)$, $x$ is a particular value of $X$, and $\varepsilon > 0$ is arbitrary, we can compute the probability
$$\mathbb{P}(|X - x| < \varepsilon) = \int_{x - \varepsilon}^{x + \varepsilon} f(x) \, dx$$
that $X$ is within $\varepsilon$ of $x$. In the continuous case, as $\varepsilon$ becomes sufficiently small this becomes asymptotic to $2 \varepsilon f(x)$, and in fact the density can be defined using only probability measurements as
$$f(x) = \lim_{\varepsilon \to 0} \frac{\mathbb{P}(|X - x| < \varepsilon)}{2 \varepsilon}.$$
This definition should remind you of a derivative, and in fact it is called the Radon-Nikodym derivative. Then we can say that the mode, if it exists, is the value of $x$ that maximizes the probability density $f(x)$, which becomes the more useful and natural substitute for the probability in the continuous case. (However, note that, unlike in the discrete case, the notion of density and hence the notion of the mode is sensitive to reparameterization.)
