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.