Back in 1907-08, William Henry Young (possibly in part, joint work with his wife, Grace Chisholm Young) proved several "nice behavior" results for arbitrary real-valued functions of a real variable.

In Theorem 6 (p. 82) of **[1]**, Young shows that for co-countably many real numbers $c$ we have

$$\liminf_{x \rightarrow c^{-}}f(x) \; = \; \liminf_{x \rightarrow c^{+}}f(x) \; \leq \; f(c) \; \leq \; \limsup_{x \rightarrow c^{-}}f(x) \; = \; \limsup_{x \rightarrow c^{+}}f(x)$$

(Note: There is a typo in the 2nd inequality at the top of p. 82: $f < {\phi}_R$ should be $f > {\phi}_{R}.)$

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).

At the 1908 International Congress of Mathematicians, Young announced (see the bottom of p. 54 of **[2]**) that for co-countably many real numbers $c,$ the left cluster set of $f$ at $c$ is equal to the right cluster set of $f$ at $c.$

This result was proved in **[3]**, where Young additionally showed that at each point of the co-countable set the value of the function belongs to the cluster set. Thus, for co-countably many real numbers $c$ we have

$$C^{-}(f,c) \; = \; C^{+}(f,c) \;\; \text{and} \;\; f(c) \in C^{+}(f,c)$$

This is a seemingly much stronger result than Young's 1907 result, since the 1907 result simply says that the endpoints of the unilateral cluster sets can only differ at countably many points, without saying anything about the distribution of the points belonging to these cluster sets.

**Definition:** Given a function $f: {\mathbb R} \rightarrow {\mathbb R}$ and $c \in {\mathbb R}$, we let $C^{+}(f,c)$ be the set of all extended real numbers $y$ (i.e. $y$ can be $-\infty$ or $+\infty$) such that there exists a sequence $\left\{x_{k}\right\}$ with each $x_k > c$ and $x_{k} \rightarrow c$ and $f(x_k) \rightarrow y.$ In other words, $C^{+}(f,c)$ is the set of all numbers (including $-\infty$ and $+\infty$) that can be obtained as a limit of $f$-values when using some sequence converging to $c$ from the right. The left version, $C^{-}(f,c),$ is defined analogously.

Regarding these results, see also Â§6 on pp. 344-346 of **[4]**.

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.

**[1]** William Henry Young, *On the distinction of right and left at points of discontinuity*, **Quarterly Journal of Pure and Applied Mathematics** 39 (1908), 67-83. [Paper dated June 1907.]

**[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.]

**[3]** William Henry Young, *Sulle due funzioni a piÃ¹ valori costituite dai limiti d'una variabile reale a destra e a sinistra di ciascun pun* [On the two functions of multiple values that are determined by the left and right limits of a real variable at each point], **Atti della Accademia Reale dei Lincei. Rendiconti. Classe di Scienze fisiche, Matematiche e Naturali** (5) 17 #9 (1st semestre) (1908), 582-587. [Paper given at session dated 3 May 1908.]

**[4]** Andrew Michael Bruckner and Brian Sheriff Thomson, *Real variable contributions of G. C. Young and W. H. Young*, **Expositiones Mathematicae** 19 #4 (2001), 337-358.

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