Zahorski theorem
Encyclopedia
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
set of zero measure.
This result was proved by Zygmunt Zahorski in 1939 and first published in 1941.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Zahorski's theorem is a theorem of real analysis
Real analysis
Real analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...
. It states that a necessary and sufficient condition for a subset of the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
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
G-delta set
In the mathematical field of topology, a Gδ set is a subset of a topological space that is a countable intersection of open sets. The notation originated in Germany with G for Gebiet meaning open set in this case and δ for Durchschnitt .The term inner limiting set is also used...
and a
set of zero measure.This result was proved by Zygmunt Zahorski in 1939 and first published in 1941.