At which points exactly has Lebesgue's singular function the derivative zero ?

Let L_a(x) be Lebesgue's singular function with a real parameter a (0<a<1, a not equal to 1/2). As is well known, L_a(x) is strictly increasing and has a derivative equal to zero almost everywhere. However, what sets of x in [0,1] actually have L_a'(x)=0 or infinity? We give a partial characterization of these sets in terms of the binary expansion of x. As an application, we consider the differentiability of the composition of Takagi's nowhere differentiable function and the inverse of Lebesgue's singular function.