Abstract: We provide an example of a one-parameter family of Cookie-Cutter-Like sets that are generated by a one-parameter family of sequences of analytic maps (varying analytically in the parameter), but for which, the Hausdorff dimension is not even differentiable as a function of the parameter. This motivates an interesting conjecture concerning the regularity of the dimension of the spectrum of a Sturmian Hamiltonian operator as a function of the coupling constant. This is a joint work with Victor Kleptsyn.