Solutions with big graph of homogeneous functional equations in a single variable (Q1271590)

Let \(X\) be a set and \(\Re\) a family of subsets of \(X\times\mathbb R\). The function \(\phi:X\to\mathbb R\) has a ``big graph'' with respect to \(\Re\) if the graph of \(\phi\) intersects every set of \(\Re\). The author offers conditions sufficient for functional equations of the form \[ g(x,\phi(x),\phi[f_{1}(x)],\phi[f_{2}(x)],\dots)=0 \] to have solutions \(\phi:X\to\mathbb R\) having big graph with respect to \(\Re\). Then properties of functions with big graphs are described. The author finishes with ``big graph solutions'' of some simple known functional equations.