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Apparently, there is a paper

M. Sablik, Final part of the answer to a Hilbert's question. Functional Equations - Results and Advances. Edited by Z. Daróczy and Zs. Páles, Kluwer Academic Publishers 2002, 231-242.

however I struggle to understand what does it have to do with Lie groups and Hilbert's fifth problem. Is it something deeper I fail to see here?

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    $\begingroup$ The paper in question does not claim to have solved Hilbert's fifth question/problem, as implied by the original title, so I have edited the title. $\endgroup$ Aug 26 '20 at 16:21
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Here's a translation of part of the address dealing with the Fifth Problem (source):

For infinite groups the investigation of the corresponding question is, I believe, also of interest. 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$^{12}$ with so much ingenuity, the difference equations, 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 the search for certain existence proofs in the calculus of variations I came directly upon the problem: To prove the differentiability of the function under consideration from the existence of a difference equation. In all these cases, then, the problem arises: In how far are the assertions which we can make in the case of differentiable functions true under proper modifications without this assumption?

The paper you link to deals with the functional equation of Abel dealt with in the paper

  • N.H. Abel, Ueber die functionen welche der Gleichung $\varphi x + \varphi y = \psi(xfy+yfx)$ genugthun, Journal für die reine und angewandte Mathematik, herausgegeben von Crelle, Bd. 2, Berlin 1827. (DigiZeit) (French translation Sur les fonctions qui satisfont à l'équation $\varphi x + \varphi y = \psi(xfy+yfx)$ available in Oeuvres complètes de Niels Henrik Abel pp 389–398, available at the Internet Archive).

The famous Cauchy functional equation $f(x+y) = f(x)+f(y)$ is a very special case, and is a baby case of the usual, finite-dimensional version of Hilbert's Fifth Problem. Or, one can view Abel's functional equation as a variant of Cauchy's with an 'exotic' addition, so in some sense a non-standard group operation on the reals. One can interpret, I gather, the group laws for a (local?) Lie group as a bunch of (coordinate) functions obeying a functional equation. Imagine you had no concept of manifold, topological space etc, but could write down what associativity meant direction in terms of coordinate functions, a priori only continuous. I would tentatively guess this is what Hilbert was driving at, allowing in the remarks above something from calculus of variations not obviously coming from groups.

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  • $\begingroup$ In a sense, the paper in the OP is not just answering a question raised as a special case of one of Hilbert's problems from 1900, but closing off a line of enquiry starting in back in 1827. $\endgroup$ Aug 27 '20 at 5:08
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Your link refers to an abstract which reads as follows:

We present new results concerning the following functional equation of Abel $$ ψ(xf(y)+yf(x))=ϕ(x)+ϕ(y) $$ D. Hilbert in the second part of his fifth problem asked whether it can be solved without differentiability assumption on the unknown functions ψ, f and ϕ. We gave earlier (cf. [9] and [10]) a positive answer assuming however that 0 is either in the domain or the range of f. Now we solve the equation in the remaining case and thus complete the answer to Hilbert’s question.

which explains, as far as I can see, how the paper in question relates to Hilbert's fifth problem; in particular, there is no claim of having solved Hilbert's fifth problem in its full sense as it is usually understood. Or is there something I fail to see?

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  • $\begingroup$ Sure, but the common understanding of Hilbert's fifth problem is that it deals with topological groups and showing that under a certain algebraic condition you only get Lie groups. I had no idea until I started looking that the original question was phrased purely in terms of functional equations and extended in this way. $\endgroup$ Aug 26 '20 at 23:20

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