A totally monotone function on a semigroup \(S\) was defined by \textit{A. Devinatz} and \textit{A. E. Nussbaum} [Duke Math. J. 28, 221-237 (1961; Zbl 0118.11201)] as a function satisfying certain difference inequalities. The author shows that the latter are equivalent to some differential inequalities if \(S\) is a Lie semigroup. This leads to some interesting properties of totally monotone functions on \(S\), in particular to a description of infinitely divisible continuous totally monotone functions, that is such totally monotone functions \(f\) that \(f(e)=1\) and \(f^{1/n}\) is totally monotone for all \(n=1,2,\ldots .\) An integral representation of totally monotone functions is also obtained.