What follows are some additional comments about this topic.
Sierpinski's 1921 paper was written without knowledge of Hahn's 1919 paper, this being a time (end of WWI) when the flow of information and journals was intermittent and/or temporarily suspended.
p. 348 in Volume 2 of Hans Hahn's Collected Works (1996) includes these remarks about Hahn's 1919 paper:
"... Hahn sets himself in Über die Menge der Konvergenzpunkte einer Funktionenfolge the task of finding out if this property gives a complete characterization of such sets. He not only proves it, but also enters the study of Baire functions by finding a characterization of the sets of convergence of functions of any given Baire class.
Jolanta Wesolowska has published versions of Hahn/Sierpinski's result for sequences of functions belonging to various other classes of functions, for example On sets of convergence points of sequences of some real functions (MR 2001d:26003; Zbl 1035.26006) and On sets determined by sequences of quasi-continuous functions (MR 2002i:26002; Zbl 1002.26004) and On sets of discrete convergence points of sequences of real functions (MR 2005f:26010; Zbl 1070.26005). Incidentally, I was in attendance when she first presented her work (represented by the first paper above) to people outside her immediate research group  (this work was her 2000 Ph.D. Dissertation at Uniwersytet Gdański), and it created a minor buzz among those attending, who found it simply amazing that the kinds of questions she was working on had not been thoroughly worked over before. (I should point out that the literature of real functions and point set theory is pretty much everywhere dense with minutia on most anything you can imagine, and more.)
 She gave this talk on 26 May 2000 at Summer Symposium in Real Analysis XXIV, held at The University of North Texas (Denton, Texas).
(Next Day) Yesterday I forgot to post a couple of references in English to a proof of the Hahn/Sierpinski result. A proof can be found on pp. 307-308 of the 1978 3rd English edition (and presumably, on the same pages for any of the other English editions) of the 1935 3rd edition of Hausdorff’s Set Theory [1957 (MR 19,111a; Zbl 81.04601); 1962 (MR 25 #4999); 1978 (Zbl 488.04001); 1991 (Zbl 896.04001); 2005] and on pp. 185-186 of Kechris’ Classical Descriptive Set Theory [MR 96e:03057; Zbl 819.04002].