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[2] William Henry Young, On some applications of semi-continuous functions, Atti del IV Congresso Internazionale dei Matematici [4th International Congress of Mathematicians] (Rome), Volume 2, 49-60. [Published version of talk given on 8 April 1908.]

[2] William Henry Young, On some applications of semi-continuous functions, Atti del IV Congresso Internazionale dei Matematici [4th International Congress of Mathematicians] (Rome), Volume 2, 49-60. [Published version of talk given on 8 April 1908.]

Another result for arbitrary real-valued functions of a real variable (it has been extensively generalized in various directions, as a google search will show) was published by Henry Blumberg in 1922, and a discussion of it can be found at the following math overflow question: Every real function has a dense set on which its restriction is continuous.

Another result for arbitrary real-valued functions of a real variable (it has been extensively generalized in various directions, as a google search will show) was published by Henry Blumberg in 1922, and a discussion of it can be found at the following math overflow question: Every real function has a dense set on which its restriction is continuous.

This implies two countability results that are now well known. The first is that for an arbitrary function, if both the left limit and the right limit exist at each point, then these unilateral limits can disagree for at most countably many points. The second is that a function can have at most countably many removable discontinuities. Incidentally, this second result was rediscovered by the Romanian mathematician Alexandru Froda in the late 1920s, and there is currently a Wikipedia page titled Froda's Theorem that is misleading at best (see Brian S. Thomson's comments here).

This implies two countability results that are now well known. The first is that for an arbitrary function, if both the left limit and the right limit exist at each point, then these unilateral limits can disagree for at most countably many points. The second is that a function can have at most countably many removable discontinuities. Incidentally, this second result was rediscovered by the Romanian mathematician Alexandru Froda in the late 1920s, and there is currently a Wikipedia page titled Froda's Theorem that is misleading at best (see Brian S. Thomson's comments here).