Example of function of Baire 3 and Baire 4 i'm looking for explicit examples of real-valued functions of the Baire third and fourth class, without using Borel-measurability but just using some characterization theorem of the previous classes. (If you know examples, please write them with the explanation of why they are in the third or fourt Baire class, please).
 A: I’m using this question as an opportunity to rescue from oblivion some little known historical comments I made in this 14 May 2009 sci.math post.
Baire $2$ and not Baire $1$
When Baire defined Baire functions in his 1899 Thesis [1], he gave an example on p. 50 of a Baire $2$ function that isn't Baire $1.$ For a discussion of this example, see the next to last paragraph of my answer to Baire class 1 and discontinuities. The example is apparently due to Volterra, according to a comment Baire makes in paragraph 38 on p. 44 of [2].
Baire $3$ and not Baire $2$
The last few pages of Baire's paper [2] (pp. 44-48) give an explicit construction of a Baire $3$ function that is not Baire $2,$ the first such example to be published. See also pp. 83, 92-97 in [3].
Baire $4$ and not Baire $3$
Keldysh gave an explicit construction of a Baire $4$ function that isn't Baire $3$ sometime in the late 1920's. The example was apparently not published by her at this time, but a lengthy discussion can be found on pp. 97-104 of [3]. Roughly 10 years later she published a short paper [4] that gave explicit examples of Baire $n$ functions that are not Baire $n-1$ for each integer $n > 1.$
[1] René Louis Baire (1874-1932), Sur les Fonctions Variables Réeles [On functions of real variables], Ph.D. dissertation (under Jean Gaston Darboux, 1842-1917), École Normale Supérieure, 1899, iv + 123 pages.

Published version: Sur les fonctions variables réeles, Annali di Matematica Pura ed Applicata (3) 3 (1899), pp. 1-123.

[2] Baire, Sur la représentation des fonctions discontinues. Première partie [On the representation of discontinuous functions. First part], Acta Mathematica 30 (1906), pp. 1-48.
[3] Nikolai Nikolaevich Luzin (1883-1950), Leçons sur les Ensembles Analytiques et leurs Applications [Lectures on Analytic Sets and their Applications], Collection de Monographies sur la Théorie des Fonctions (Borel’s
Series) #37?, Gauthier-Villars, 1930, xvi + 328 pages. Reprinted in 1972 by Chelsea Publishing Company.
[4] Lyudmila Vsevolodovna Keldysh (1904-1976), Démonstration directe du théorème sur l’appartenance d’un élément canonique $E_{\alpha}$ à la classe $\alpha$ et exemples arithmétiques d’ensembles mesurables $B$ de classes supérieures [A direct proof of a theorem that the canonical element $E_{\alpha}$ belongs to the class $\alpha,$ and arithmetic examples of $B$ sets of high classes], Doklady Akademii Nauk SSSR [= Comptes Rendus de l’Académie des Sciences de l'URSS] (N.S.) 28 #8 (1940), pp. 675-677.
