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In mathematics, Zahorski's theorem is a theorem of real analysis. It states that a necessary and sufficient condition for a subset of the real line to be the set of points of non-differentiability of a continuous real-valued function, is that it be the union of a Gδ set and a \( {\displaystyle {G_{\delta }}_{\sigma }} \) set of zero measure.

This result was proved by Zygmunt Zahorski [pl] in 1939 and first published in 1941.


Zahorski, Zygmunt (1941), "Punktmengen, in welchen eine stetige Funktion nicht differenzierbar ist", Rec. Math. (Mat. Sbornik) N.S. (in Russian and German), 9 (51): 487–510, MR 0004869.
Zahorski, Zygmunt (1946), "Sur l'ensemble des points de non-dérivabilité d'une fonction continue" (French translation of 1941 Russian paper), Bulletin de la Société Mathématique de France, 74: 147–178, MR 0022592.

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