We consider asymptotic values of continuous functions u from $\mathbb{R}^m$, or a suitable subset of $\mathbb{R}^m$, to the extended real numbers $\overline\mathbb{R}\,{=}\,\mathbb{R}\,{\cup}\,\{-\infty\}\,{\cup}\,\{\infty\}$, with the usual topologies in each case. Building on work of Rippon we obtain some rather unexpected results about such asymptotic values, in particular an intermediate value property.