Some question regarding how probability is defined Consider an arbitrary discrete probability distribution with sample space $\Omega$ and let $\omega\subset\Omega$. Let $n$ denote the amount of independent trials of an experiment that are performed and let $\operatorname{f}(n)$ equal the amount of times $\omega$ occurs during those $n$ trials. 
It is my understanding that  $\operatorname{P}(\omega)=\lim_{n\to\infty}\operatorname{f}(n)/n$. Is $\operatorname{f}$ essentially "pure randomness"? I mean we can't necessarily be certain about what value we acquire from $\operatorname{f}$ when evaluated at $n$. I'm used to a function giving me the same number when I iteratively evaluate it at the same number, but this isn't the case now is it? Does it make sense for $\operatorname{f}$ to exist philosophically? 
If "pure randomness" determines the value of $\operatorname{f}(n)$ in the sense that we can never be $100$% certain about we value it will yield, how do we define "pure randomness"? 
Since $\operatorname{f}$ is not a normal function like those in calculus, how do we define the convergence of $\operatorname{P}(\omega)=\lim_{n\to\infty}\operatorname{f}(n)/n$? Does the $\delta$-$\epsilon$ kind of definition apply here as well? How rigorous is this definition generally speaking? 
In addition to that, how do we define probability for continuous probability distribution in a more rigorous way?
 A: To be specific, toss a coin repeatedly, and let $X_n$ be the number of
Heads in $n$ tosses. According to the frequentist 'definition' of
probability, we say that $P(\text{Heads}) = 1/2$ (coin is 'fair') if
$R_n = X_n/n$ "approaches" 1/2 with increasing $n$.
But, you're right, this use of "approaches" cannot refer a limit in the same traditional mathematical sense that
the deterministic sequence $A_n = (1 + \frac{1}{n})^n$ approaches $e$.
We know in advance precisely the value of the deterministic sequence $A_n$ for
each $n.$ And we cannot know the value of $R_n$ for each $n$ without
doing the "random" coin tosses.
One way to make the random situation rigorous is to say that $R_n$ converges 'in probability' to 1/2 with
increasing $n.$ This is defined as saying that $Q_{n,\epsilon}$ converges
to $1$ in the traditional mathematical sense, for any $\epsilon > 0$, where
$Q_{n,\epsilon}$ is defined by
$$Q_{n,\epsilon} = P(|R_n - 1/2| < \epsilon).$$
You might see such notations as "$\text{plim}\, R_n = 1/2$" or
"$R_n \stackrel{prob}{\rightarrow} 1/2.$"
Once you have learned a bit more probability, you will recognize that
$X_n$ is a 'binomial random variable' with $n$ independent trials and
'Success probability' 1/2 at each trial. Then for any given positive
$\epsilon,$ the quantity $Q_{n,\epsilon}$ has a known value in advance.
This limiting relationship is a special case of the '(Weak) Law of Large Numbers',
which I suppose you will study in due course.
To illustrate, below is a graph of the actual values of $Q_{n,\epsilon},$ for
$n = 1, 2, \dots, 800$ and $\epsilon = 0.05,$ made using R statistical
software.
n = 1:800;  eps=.05
Q = pbinom(n*(.5+eps), n, .5) - pbinom(n*(.5-eps)+.00001, n, .5)
plot(n, Q, type="l");  abline(h=1, col="darkgreen")


Note: You do not mention the mathematical level of your course. My
explanation is intuitive and I hope appropriate for the beginning of an undergraduate
post-calculus course in probability. If you are studying measure theoretic
probability, then please refer to @Did's more rigorous and elegant approach.
A: Here is a more rigorous measure-theoretic treatment of the problem.
Consider the space $\Omega^\mathbb{N}$ equipped with the product measure. You can think of this as the space of infinite sequences of outcomes in $\Omega$, i.e. the result of performing a random experiment infinitely many times. Now, we can define $X_n$ to be the indicator function of $\pi_n^{-1}(A)$ - here $\pi_n$ is the function mapping a point to its $n$th coordinate, so $X_n$ is $1$ whenever the $n$th coordinate lies in $A$, and $0$ otherwise. Then the random variable $Y_n = \sum^n_{i=1} X_i$ represents the number of times $A$ occurs in $n$ trials. Note that $Y_n$ is not a number, but rather a function from $\Omega^\mathbb{N}$ to $\mathbb{R}$. (This is why your notation of $f(n)$ is misleading, as it suggests that $f$ depends only on $n$, when in fact it is a function of the sample space as well). Thus, for each point in $\Omega^\mathbb{N}$, we can talk about $\frac{Y_n}{n}$ as a sequence of real numbers, and determine its limit, if it converges at all. A special case of the Strong Law of Large Numbers then states that $\frac{Y_n}{n} \rightarrow P(A)$ almost surely (the set of points where it doesn't converge to $A$ has probability $0$). The application of this law requires that the $X_n$ be independent and identically distributed, which should be intuitively obvious since each one depends only on the $n$th coordinate, which is independent of all the others.
Sidenote: you'll notice I used $A$ instead of $\omega$ to denote the set. This is because in probability theory, typically capital letters like $A$,$B$, etc. are used to denote subsets of the sample space, whereas $\omega$ denotes an element of the sample space.
