  Given a general channel, we first formulate the idetification capacity
problem as well as the resolvability problem with input cost constraint in as
the general form as possible, along with relevant fundamental theorems. Next,
we establish some mild sufficient condition for the key lemma linking the
identification capacity with the resolvability to hold for the continuous input
alphabet case with input cost constraint. Under this mild condition, it is
shown that we can reach the {\em continuous}-input fundamental theorem of the
same form as that for the fundamental theorem with {\em finite} input alphabet.
Finally, as important examples of this continuous-input fundamental theorem, we
show that the identification capacity as well as the resolvability coincides
with the channel capacity for stationary additive white (and also non-white)
Gaussian noise channels.
