Applications of divergence points to local dimension functions of subsets of $\Bbb R^{d}$

For a subset E subset of or equal to R-d and x is an element of R-d, the local Hausdorff dimension function of E at x is defined by dim(loc) (x, E) =(rSE arrow 0) lim dim (E boolean AND B (x, r)),

where 'dim' denotes the Hausdorff dimension. Using some of our earlier results on so-called multifractal divergence points we give a short proof of the following result: any continuous function f : R-d -->4 [0, d] is the local dimension function of some set E subset of or equal to R-d. In fact, our result also provides information about the rate at which the dimension dim(E boolean AND B(x, r)) converges to f (x) as r SE arrow 0.