  A method for computing low--temperature series for renormalized operators in
the two--dimensional Ising model is proposed. These series are applied to the
study of the properties of the truncated renormalized Hamiltonians when we
start at very low temperature and zero field. The truncated Hamiltonians for
majority rule, Kadanoff transformation and decimation for $2 \times 2$ blocks
depend on the how we approach the first--order phase--transition line. These
Renormalization Group transformations are multi--valued and discontinuous at
this first--order transition line when restricted to some finite--dimensional
interaction space.
