I will discuss the question of regularity of the function ascribing to a parameter the numerical value of the Hausdorff measure of the limit set for some naturally parametrized families of conformal dynamical systems. Holder continuity of this function will be proved for several natural classes of parametrized families.

I will also present some results on continuity of Hausdorff measure function in the context of natural conformal dynamical systems related to continued fraction expansion.