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A new type of fractal measures Xs 1 < s < 2, defined on the subsets of the graph of a continuous function is introduced. The J-dimension defined by this measure is 'closer' to the Hausdorff dimension than the other fractal dimensions in recent literatures. For the Weierstrass type functions defined by W(x) = A-aig(Aix), where A > 1, 0 < a < 1, and g is an almost periodic Lipschitz function of order greater than a, it is shown that the %-dimension of the graph of W equals to 2 a, this conclusion is also equivalent to certain rate of the local oscillation of the function. Some problems on the 'knot' points and the nondifferentiability of W are also discussed.

Fractal Dimensions and Singularities of the Weierstrass Type Functions