As, beginning with the famous Hofstadter's butterfly, all
numerical studies of spectral and dynamical quantities related to
quasiperiodic operators are actually performed for their rational
frequency approximants, the questions of continuity upon such
approximation are of fundamental importance. The fact that continuity
issues may be delicate is illustrated by the recently discovered
discontinuity of the Lyapunov exponent for non-analytic potentials.

I will review the subject and then focus on work in progress, joint with Avila and Sadel, where we develop a new approach to continuity, powerful enough to handle matrices of any size and leading to a number of strong consequences.

This is the first of (likely) two talks, where an almost entire proof will be presented. For understanding most of the talk knowledge of spectral theory should not be necessary and just knowing some basic harmonic analysis should suffice.