  We study the classical problem of characterizing the channel capacity and its
achieving distribution in a generic fashion. We derive a simple relation
between three parameters: the input-output function, the input cost function
and the noise probability density function, one which dictates the type of the
optimal input. In Layman terms we prove that the support of the optimal input
is bounded whenever the cost grows faster than a cut-off rate equal to the
logarithm of the noise PDF evaluated at the input-output function. Furthermore,
we prove a converse statement that says whenever the cost grows slower than the
cut-off rate, the optimal input has necessarily an unbounded support. In
addition, we show how the discreteness of the optimal input is guaranteed
whenever the triplet satisfy some analyticity properties. We argue that a
suitable cost function to be imposed on the channel input is one that grows
similarly to the cut-off rate. Our results are valid for any cost function that
is super-logarithmic. They summarize a large number of previous channel
capacity results and give new ones for a wide range of communication channel
models, such as Gaussian mixtures, generalized-Gaussians and heavy-tailed noise
models, that we state along with numerical computations.
