  We introduce a polydisperse version of the streaming instability, where the
dust component is treated as a continuum of sizes. We show that its behaviour
is remarkably different from the monodisperse streaming instability. We focus
on tightly coupled particles in the terminal velocity approximation and show
that unstable modes that grow exponentially on a dynamical time scale exist.
However, for dust to gas ratios much smaller than unity they are confined to
radial wave numbers that are a factor $\sim 1/\overline{\rm St}$ larger than
where the monodisperse streaming instability growth rates peak. Here
$\overline{\rm St} \ll 1$ is a suitable average Stokes number for the dust size
distribution. For dust to gas ratios larger than unity, polydisperse modes that
grow on a dynamical time scale are found as well, similar as for the
monodisperse streaming instability and at similarly large wave numbers. At
smaller wave numbers, where the classical monodisperse streaming instability
shows secular growth, no growing polydisperse modes are found under the
terminal velocity approximation. Outside the region of validity for the
terminal velocity approximation, we have found unstable epicyclic modes that
grow on $\sim 10^4$ dynamical time scales.
