  We prove that the graph of a discontinuous $n$-monomial function
$f:\mathbb{R}\to\mathbb{R}$ is either connected or totally disconnected.
Furthermore, the discontinuous monomial functions with connected graph are
characterized as those satisfying a certain big graph property. Finally, the
connectedness properties of the graphs of additive functions
$f:\mathbb{R}^d\to\mathbb{R}$ are studied.
