On functions whose graph is a Hamel basis

Abstract:

We say that a function $h \colon \mathbb {R} \to \mathbb {R}$ is a Hamel function ($h \in \textrm {HF}$) if $h$, considered as a subset of $\mathbb {R}^2$, is a Hamel basis for $\mathbb {R}^2$. We prove that every function from $\mathbb {R}$ into $\mathbb {R}$ can be represented as a pointwise sum of two Hamel functions. The latter is equivalent to the statement: for all $f_1,f_2 \in \mathbb {R}^{\mathbb {R}}$ there is a $g\in \mathbb {R}^{\mathbb {R}}$ such that $g+f_1,g+f_2\in \mathrm {HF}$. We show that this fails for infinitely many functions.