Consider a function $mu(s)$ satisfying the following properties:

- $mu(s) in C^0((0,+infty))$,
- $mu(s) > 0$ and $mu(s)$ is increasing in $s in (0,+infty)$,
- $displaystyle int_0^1 {dfrac{mu(s)}{s} ds} < +infty.$

Show that for every $epsilon > 0$, there is a $delta >0 $ such that$$displaystyle int_0^delta {dfrac{mu(s)}{s} ds} < epsilon.$$

**My idea:** Set $displaystyle F(delta) := int_0^delta {dfrac{mu(s)}{s} ds}$. Then the conclusion follows if we can show that

$$ lim_{delta to 0^+} {F(delta)} = 0 $$This, in turn, will follow if $F$ is continuous on $[0,delta)$. There will be no problem if $mu(s) /s$ were bounded on $[0,delta)$ but this is not always the case; take $mu(s) = s^a (0