# Real Analysis – A continuous and growing function with special scaling and integral properties

I would like to get an explicit description of a continuous function, not decreasing $$f: (0,1) to (0, infty)$$ with the following properties: (i) $$f (0) = 0$$; (Ii) $$int_0 ^ 1 f (x) / x dx = infty$$ and (iii) $$liminf_ {x to 0} f (x ^ 2) / f (x) = 0$$.

Any help appreciated.