Algebraic properties of Hamel functions

We say that f: ℝ → ℝ is LIF if it is linearly independent over ℚ as a subset of ℝ² and that it is a Hamel function (HF) if it is a Hamel basis of ℝ². We construct an example of HF bijection and use a similar method to prove that any function can be represented as the composition of three HF’s as well as the limit of uniformly convergent sequence of HF’s. Finally we consider products of HF’s, maximal invariant classes (with respect to several algebraic operations) and pose some open problems concerning sets of continuity points of HF’s.