In Uni, we learned about the Landau-symbol $\mathcal{O}$. But the definition confuses me.
$f=\mathcal{O}(g)$ for $x\rightarrow x_0$ if and only if there exists a constant $c>0$ and a neighbourhood $U(x_0)$, so that for every $x \in U(x_o): |f(x)| \le c\cdot|g(x)|$
Now I was thinking that this counts for almost every function. E.g. let's take $f(x) = x^2$ and $f(x)=x$.
Let's also choose a neighbourhood: $U(x_0) = [-10,10]$ for $x_0=0$. Obviously for any $x\in U(x_0)$, I could say that $|x^2| \le c\cdot|x|$ with $c = 10$. My problem is that for any neighbourhood, I could choose a constant c that is high enough to satisfy the criterion.
Isn't the criterion always satisfied simply if c is large enough?