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Arbitrarily slow approach to limiting behavior

Let f(k,t):R N ×[0,∞)→R be jointly continuous in k and t, with lim t→ ∞f(k,t)=F(k) discontinuous for a dense set of k's. It is proven that there exists a dense set Γ of k's such that, for k∈Γ, |f(k,t)−F(k)| approaches 0 arbitrarily slowly, i.e., more slowly than any expressible function g(t)→0. The result is applied to diffusion and conduction in quasiperiodic media and yields arbitrarily slow approaches to limiting behavior as time or volume becomes infinite. Such a slow approach is in marked contrast to the power laws widely found for random media