Let O(U) denote the finely harmonic functions on U a finely open subset of C such that ∂g ∂ z ̄ = 0 almost surely on U. Define Af(K) to be those g in C(K) such that if K′ is the fine interior of K then g |K′ is in O(K). We prove that Af(K) is invariant under the Vitushkin localization operators, i.e., it is T-invariant. We also settle an open question of Fuglede on the existence of polygonal arcs in finely open subsets of Rn. Using T-invariance we prove that point derivation yields a continuo...