Question 129
Let $x_0$ be a real number. Prove that there exists a useful function $f$ so that $f(x_0) = 0$ if and only if $|x_0| \ge 4$.
We will call an affine function $f : \mathbb{R} \to \mathbb{R}$ *useful* if it has the properties:
(i) $|f(x)| \le 2$, for every real number $x$ such that $|x| \le 2$;
(ii) $|f(x)| \ge 1$, for every real number $x$ such that $|x| \le 1$.
We will call an affine function $f : \mathbb{R} \to \mathbb{R}$ *useful* if it has the properties:
(i) $|f(x)| \le 2$, for every real number $x$ such that $|x| \le 2$;
(ii) $|f(x)| \ge 1$, for every real number $x$ such that $|x| \le 1$.