The group of diffeomorphisms of a non compact manifold is not regular
  We show that a group of diffeomorphisms $\D$ on the open unit interval $I,$ equipped with the topology of uniform convergence on any compact set of the derivatives at any order, is non regular: the exponential map is not defined for some path of the Lie algebra. this result extends to the group of diffeomorphisms of finite dimensional, non compact manifold $M.$ 
Jean-Pierre Magnot
math.DG