Frederick R. Adler, and Julio Mosquera. 2000. Is space necessary? Interference competition and limits to biodiversity. Ecology 81:3226-3232.
To prove that an analytic function b cannot support a stable coalition with a continuum of species, we require the following two lemmas. First, we assume that the coalition p(m) has ``compact support", meaning that the mortality rates are bounded above, and that it has a finite integral (as it must). Then p(m) is an L1 function. The key result is that if b is an analytic function, the convolution of p with b will also be analytic, which follows from Morera's Theorem (Rudin, 1974).
Given this lemma, we have that f(m) is analytic if b is. The same must hold for the second derivative of f. If the coalition included a continuum of species, the second derivative would have to be zero on an interval. However, an analytic function cannot be exactly 0 on an interval without itself being identically zero everywhere. This is only possible if the second derivative of b is zero, or if b(z) is a linear function. Such a competitiveness function cannot be bounded above. Furthermore, solutions with this competitiveness function are degenerate, supporting an infinite number of neutrally stable coalitions (results not shown).
Rudin, W. 1974. Real and Complex Analysis. McGraw-Hill, Inc., New York, New York, USA.