A function whose graph is of dimension 1 and has locally an infinite one-dimensional Hausdorff measure

P. Wingren [4] asked whether there exists a real-valued continuous nowhere differentiable function on [0, 1] with the following properties: - the Hausdorff dimension of its graph equals 1; - if the projection on [0, 1] of a subset of its graph has a positive Lebesgue measure, then then its one-dimensional Hausdorff measure is infinite. We construct a function fulfilling these requirements. Moreover, we also find the gauge function of its graph. (C) 2001 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.