Asymptotic behaviour and bound for function

Question

Suppose we have a continuous postive function $f(t)$ (if you need some extra regularity, you can assume it) such that $f$ is non decrasing, so for all $t_2 > t_1$ we have \begin{align*} f(t_2) \geq f(t_1) \end{align*} and we also know the asymptotic behaviour, in particular \begin{align*} f(t) \sim t \quad t \to +\infty \end{align*} so we can say that \begin{align*} \lim_{t \to +\infty} \frac{f(t)}{t}=c \end{align*} where $c>0$ is a positive constant. Now we want to find a bound for $f(t)$, for example we can say that $\exists M>0$ such that \begin{align*} f(t) \leq M t \quad \forall t \end{align*} or we can find another bound for $f(t)$ such that $f(t) \leq ... t ...$ for all t? Do you have any ideas for a bound for $f(t)$ ? Thanks

Answer 1

There is a $t_0$ such that $f(t) \leq (c+1) t$ for $t \geq t_0$. But your inequality need not hold for $t$ near $0$: Take $f(x)=\sqrt x$ for $0 <x<1$ and $f(x)=x$ for $x \geq 1$.