Why is $G(0) = 0 $?

Question

So the task is the following:

Suppose $f : x ∈ R \rightarrow R$, $f(0) = 0$ and $f$ is continous on 0. Let $g : R \rightarrow R$ be so that $|g(x)| \leq |f(x)|$ $\forall x∈R$. Show that $g$ is continous on 0 by the $\epsilon-\delta$-definition.

As far as I understand $g(0)$ need to be equal to $0$ for this to work. But what I don't understand is how we definitely can know that $|g(0)| = |f(0)|= 0$, when we only know that $|g(0)| \leq |f(0)| = 0$?

Answer 1

Hint: $|x| \le 0 \iff x = 0$ since $| \cdot | \ge 0$.