Positive functions which doesn't get multiplied too fast/doesn't grow too fast What could be some non-trivial examples of positive valued continuous functions $f: \mathbb{R_{+}}\to \mathbb{R_{+}}$ so that:
$\forall x, y \in \mathbb{R_{+}}, x<y$, $f(x) \geq \frac{1}{2}f(y)$?
Of course, the constant and monotone non-increasing functions satisfy this property trivially. Also we note that locally,  when $x, y$ are close enough, this property is valid, since for close enough $x,y$, $f(x)$ is approximately equal to $f(y)$, hence $f(x) \geq \frac{1}{2}f(y)$. I'm looking for other examples, or rather a class of examples, or a way to construct the examples globally. See below.
For a non-constant example, one can define: $f: (0,\frac{\mu}{\lambda})\to \mathbb{R_{+}}$ by: $f(x)= \lambda x + \mu$, and this will satisfy the desired property. But $f$ won't satisfy the desired property when $x > \frac{\mu}{\lambda}$, so the definition of $f$ needs to be modified on $[\frac{\mu}{\lambda}, \infty)$. 

This is achieved if $M\le2m$ with $m:=\inf S,\,M:=\sup S$, with $S$ the set of values of $f$. So take your favourite bounded function, and linearly transform it to any $m,\,M$ of your choosing with $M\le2m$. If we first transform so $f$ ranges from $0$ to $1$ (e.g. $f=\frac{2}{\pi}\arctan x$), $f$ doesn't work, but $f+c$ does for any $c\ge1$.
