  We prove almost Lipshitz continuity of spectra of singular quasiperiodic
Jacobi matrices and obtain a representation of the critical almost Mathieu
family that has a singularity. This allows us to prove that the Hausdorff
dimension of its spectrum is not larger than 1/2 for all irrational
frequencies, solving a long-standing problem. Other corollaries include two
very elementary proofs of zero measure of the spectrum (Problem 5 in [41]) and
a similar Hausdorff dimension result for the quantum graph graphene.
