Continuity of delta invariants and twisted K\"ahler--Einstein metrics

We show that delta invariant is a continuous function on the big cone. We will also introduce an analytic delta invariant and show its continuity in the K"ahler cone, from which we deduce the continuity of the greatest Ricci lower bound. Then building on the work Berman-Boucksom-Jonsson, we obtain a uniform Yau-Tian-Donaldson theorem for twisted K"ahler-Einstein metrics in general K"ahler classes.