Let $f:[0, r] \to [0, \infty)$ be continuous, positive definite, and increasing function, i.e., it is continuous and
1) $f(0) = 0$ and $f(x) > 0$ for $x \in (0, r]$;
2) $f(x) \leq f(y)$ whenever $x < y$ for $x, y \in [0, r]$.
Then, I want to show that there exist continuous strictly increasing functions $f_1:[0,r] \to [0, \infty]$ and $f_2:[0,r] \to [0, \infty]$ such that
1) $f_1(0)=f_2(0) = 0$
2) $0 < f_1(x) \leq f(x) \leq f_2(x)$ for all $x \in (0, r]$.
I guess that such functions always exist for any function $f$ satisfying the properties, but stuck with the proof.