HAUSDORFF-BESICOVITCH DIMENSION OF THE GRAPH OF ONE CONTINUOUS NOWHERE-DIFFERENTIABLE FUNCTION

We investigate fractal properties of the graph of the function y = f(x) = Sigma(infinity)(k=1) beta(k)/2(k) Delta(2)(beta 1 beta 2...beta k...), where beta(1) = {0 if alpha(1) (x) = 0, 1 if alpha(1) (x) not equal 0, beta k = {beta(k-1) if alpha(k)(x) = alpha(k-1)(x), 1 - beta(k-1) if alpha(k)(x) not equal alpha(k-1)(x), k > 1, and alpha(k)(x) is the kth ternary digit of x. In particular, we prove that this graph is a fractal set with Hausdorff-Besicovitch dimension alpha(0)(Gamma(f)) = log(2) (1 + 2(log32)) and cell dimension alpha(K)(Gamma(f)) = 2 - log(3)2.