Functional equations generating space-densifying curves

In this paper, infinite-dimensional vector spaces of α-dense curves are generated by means of the functional equations f(x) + f(2x) = 0 and f(x) + f(2x) + f(3x) = 0. For both equations, a basis of continuous solutions has been constructed, obtaining in this way the main achievement which is the fact of that these bases may α-densify a large class of compact sets of the plane for arbitrary small α. Finally, a constructive method to find the noncontinuous solutions of the second functional equation is also settled.