  Totally symmetric continuous spin field propagating in (A)dS is studied.
Lagrangian gauge invariant formulation for such field is developed. Lagrangian
of continuous spin field is constructed in terms of double traceless tensor
fields, while gauge transformations are constructed in terms of traceless gauge
transformation parameters. de Donder like gauge condition that leads to simple
gauge fixed Lagrangian is found. Gauge-fixed Lagrangian invariant under global
BRST transformations is presented. The BRST Lagrangian is used for computation
of a partition function. It is demonstrated that the partition function of the
continuous spin field is equal to one. Various decoupling limits of the
continuous spin field are also studied.
