On functions whose graph is a Hamel basis

Abstract

We say that a function h: R→ R is a Hamel function (h E HF) if h, considered as a subset of R 2 , is a Hamel basis for R 2 . We prove that every function from R into R can be represented as a pointwise -sum of two Hamel functions. The latter is equivalent to the statement: for all f 1 , f 2 ∈ R R there is a g ∈ R R such that g+f 1 , g + f 2 ∈ HF. We show that this fails for infinitely many functions.