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Let L_a(x) be Lebesgue's singular function with a real parameter a (0<a<1, anot equal to 1/2). As is well known, L_a(x) is strictly increasing and has aderivative 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 ofthese sets in terms of the binary expansion of x. As an application, weconsider the differentiability of the composition of Takagi's nowheredifferentiable function and the inverse of Lebesgue's singular function.

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