A construction is put forward which shows how to decompose an arbitrary continuous function f ∈ C[0, 1] into a sum of two continuous functions each having a graph of Hausdorff dimension one. An example is given where the terms in the decomposition have graphs of Hausdorff dimension 1 and packing dimension 2.

Abstract. A class of subsets of R n is constructed that have certain homogeneity and non-coincidence properties with respect to Hausdorff and box dimensions. For each triple (r, s, t) of numbers in the interval (0, n] with r < s < t, a compact set K is constructed so that for any non-empty subset U relatively open in K, we have

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A method to decompose real valued continuous functions defined on R is put forward. The decomposition is by infinite linear combinations of B-splines. It is proved that a necessary and sufficient condition for a function to be in the Zygmund space is that the corresponding sequence of coefficients be in the sequence space l . Academic Press