  We prove that norm continuous derivations from a von Neumann algebra into the
algebra of operators affiliated with its tensor square are automatically
continuous for both the strong operator topology and the measure topology.
Furthermore, we prove that the first continuous L^2-Betti number scales
quadratically when passing to corner algebras and derive an upper bound given
by Shen's generator invariant. This, in turn, yields vanishing of the first
continuous L^2-Betti number for II_1 factors with property (T), for finitely
generated factors with non-trivial fundamental group and for factors with
property Gamma.
