We study continuous transformations preserving the Hausdorff - Besicovitch dimension ("DP-transformations") of every subset of <$E {bold roman R} sup 1>. It is shown that the problem of investigation of continuous DP-transformations of the real line is equivalent to the problem of studying the DP-properties of strictly increasing probability distribution functions on a unit interval. Applying the multilevel fractal analysis of singularly continuous probability measures with independent Q-digits, we found sharp (necessary and sufficient) conditions for the Hausdorff - Besicovitch dimension preservation under the corresponding distribution functions.