In this article we discuss the requirements needed in order to characterise the solution space of perturbed linear integro-differential Volterra convolution equations. We highlight how in general the pointwise behaviour of perturbation functions does not necessarily propagate through to the solution which the classical literature seems to suggest. To illustrate this general idea we show the Cesàro mean of the solution can converge even in cases when the Cesàro mean of the perturbation function diverges. We provide a characterisation of when such convergence takes place and explicitly identify the limit in terms of the problem data. Furthermore we prove how all results can also be applied to perturbed linear functional differential equations.