The mass-insertion term in the Callan-Symanzik equation may not be negligible, even in the deep Euclidean region, because the asymptotic behavior of vertex functions may be different from that of individual dominant graphs. When contributions from different orders of perturbation are related by a constraint, such as the condition that the physical coupling constant is at a nontrivial Callan-Symanzik eigenvalue, cancellations between dominant graphs may occur. This and other circumstances under which vertex functions may fail to satisfy homogeneous Callan-Symanzik equations asymptotically are illustrated with a very simple example, thus emphasizing that the question of asymptotic scale invariance may not be always decided by studying the properties of the $\ensuremath{\beta}$ function alone.