Divergence-example of a non-negative function with $\lim_{x\to0} f(x)=0$ and $\int^1_0 \frac{f(x)}{x} dx = \infty$

Question

I want to know an example of a real valued function $f \in C([0,1])$ satisfying

$f(x) \geq 0 \ \forall x \in [0,1]$,

$\lim_{x \to 0} f(x) = 0$, and

$\int^1_0 \frac{f(x)}{x} dx = \infty$.

Any advice would be appreciated.

Answer 1

Hint. Try with $$f(x)=\frac{1}{|\ln(x/2)|^a}$$ with some $a>0$ (extended at $0$ with its limit).