In a discrete-time continuous-alphabet memoryless channel we impose an average cost constraint $(t(\cdot),P)$ on any input $n$-tuple $x^{n}=(x_{1},\ldots,x_{n})$ sent over the channel by requiring that $$\frac{1}{n}\sum\limits_{i=1}^{n}t(x_{i})\le P$$where $t:\mathcal{X}\to[0,\infty)$ is an input cost function and $P>0$ denotes the budget per input symbol (i.e. the maximal average cost allowed per input symbol).