Problem: Let $f : \mathbb{R}\times \mathbb{R}\to\mathbb{R}$ be a function that is not continuous. Let's say the function terms would be $x$ and $y$, so that we'll have $f(x, y)$. How could I determine which of the two terms affects the function value more?
Mention: As the function is not continuous, I believe we cannot compute the partial derivatives of the function. (Might be wrong)
Mention 2: Basically, I believe what I'm trying to do is to determine the Big-O for the given function.
Big $O$ is about comparing the behavior of two functions near a point. It does not require either of the two functions to be continuous. You talk about finding "the" big $O$ as if for any function, there was only 1 possible comparison. But that is not the case. There are an infinite multitude of functions to which your function can be compared.
Certainly if $f \in O(g)$ and $g \in O(h)$, then $f \in O(h)$ as well. But there are also plenty of function $u, v$ such that $u \notin O(v)$ and $v \notin O(u)$, yet $f \in O(u)$ and $f \in O(v)$.
For $f$ to be $O(g)$ (at some point $a$ - usually $0$ or $\infty$ and determined by context), it means there is a neighborhood of $a$ and a constant $M > 0$ such that for all $x$ in the neighborhood, $|f(x)| \le Mg(x)$.
The method for finding a $g$ that is adequate for your needs is going to depend first on what those needs are, and second on what you know about $f$. If you know $f$ is differentiable, then the derivatives give polynomial approximations. When $f$ is not continuous, you have to resort to other tactics. What those tactics are changes from case to case. There isn't some algorithm that will handle them all. But all such tactics require you to know something about the behavior of the function. If you have no idea what the function is doing, then you are out of luck. You can't pull a rabbit out of an empty hat.