The State of the Second Part of Hilbert's Fifth Problem

"Moreover, we are thus led to the wide and interesting field of functional equations which have been heretofore investigated usually only under the assumption of the differentiability of the functions involved. In particular the functional equations treated by Abel (Oeuvres, vol. 1, pp. 1,61, 389) with so much ingenuity...and other equations occurring in the literature of mathematics, do not directly involve anything which necessitates the requirement of the differentiability of the accompanying functions... In all these cases, then, the problem arises: In how far are the assertions which we can make in the case of differentiatie functions true under proper modifications without this assumption?" (Hilbert's emphasis).