An upper bound for $f$, given a differential inequality Let $f : \mathbb R \to (0, \infty)$ be a function (At least a locally Lipschitz function) so that 
$$f + f' \leq C$$
for some positive constant $C$. Does that imply $f\leq C'$ for some $C'$? 
Of course $f\leq C$ on the set $\{ f' \geq 0\}$ and that $f$ "cannot increase" on the set $\{ f' <0\}$. But I have problem making that rigorous. Any help appreciated!

Define $g(x) = f(x)e^{x}$ then 
$$f+f' \leq C \implies g' \leq Ce^{x} \implies f(x) \leq C + De^{-x}$$
for some (integration) constant $D$ (which can be expressed as $D = f(0) - C$).
