AbstractLet O(U) denote the finely harmonic functions on U a finely open subset of C such that ∂g∂z̄ = 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 continuous functional on Af(K) for each point of K′. Using the polygonal arc results as well one can show that for a large class of z in K′, f(ω) = (g(ω) − g(z))(ω − z) extends across z and is in O(K), from which f(z) = g′(z). We also establish that R(K) and Af(K) have the same Arens-Singer and Jensen measures.