Landau symbol definition

Question

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?

Answer 1

We cannot have $c$ such that $|x| \leq c|x^{2}|$ for $x$ in some interval around $0$. This is because the inequality fails whenever $0<|x| <\frac 1 c$.

Answer 2

No. Let $f(x)=x, g(x)=x^2$ and $x_0=0.$ Suppose that there is $c>0$ and a neigborhood $U$ of $x_0$ such that $|f(x)| \le c\cdot|g(x)|$ for all $x \in U$. This would imply that

$1 \le c|x|$ for all $x \in U$ with $x \ne 0.$

But this is absurd.