I am self studying analysis and came across a theorem that states that for any real valued function $f$, the set of points where the upper right derivative of $f$ is strictly less than the lower left derivative of $f$ is countable set i.e. $$D^+f(x) < D_-f(x)$$ is a countable set. Unfortunately, no such proof was given for this theorem. Could someone provide a proof or direct me to where I could find one so I could get better insight?
Here are the definitions that I am working with. Let $f$ be a real valued function on $[a,b]$ and let $x_0 \in [a,b]$. Then $$D^+ f(x_0) = \limsup_{x \to x_0+} \frac{f(x) - f(x_0)}{x - x_0}$$ and $$D_- f(x_0) = \liminf_{x \to x_0-} \frac{f(x) - f(x_0)}{x - x_0}$$
As Dave Renfro points out in his comment, the Young family in the UK back in the early years of the 20th century studied problems of this type--asymmetry between right and left, in various contexts such as cluster sets or Dini derivatives.
But the credit for this one goes to Grace, not William.
She is justly famous for a later, deeper study of this type known now as the Denjoy-Young-Saks theorem, also about the disposition of the four Dini derivatives for a general function.
A note on derivates and differential coefficients. Grace Chisholm Young. Acta Math. 37: 141-154 (1914)
She credits the idea, however, to the Italian mathematician Beppo Levi. [I love to cite this guy and always think of him as somehow a member of the Marx family.]
Richerce sulle funzioni derivate. Beppo Levi. Rend. dei Lincei. 1906
Beppo proved the weaker version: The set of points where the right-hand derivative and the left-hand derivative of a function both exist but are different is at most countable.
Advice: Prove Beppo's theorem as a warm-up to that of Grace. Same method will work but you might find limsups and liminfs and right/left a bit intimidating to start with. Just show that the set $\{x: f'_+(x)< f'_-(x)\}$ is countable. Hint: Show that the set $\{x: f'_+(x)<r< f'_-(x)\}$ is countable for each rational number $r$.
References [added 1/16/2022]
[1] S. Saks, Theory of the Integral, 1937. https://eudml.org/doc/219302
Indispensible reference for all real analysis students. Grace Young's theorem appears in Theorem 1.1, Chapter IX, p. 261 with a very simple proof. There is much more information about the Dini derivatives in earlier chapters.
[2] A. M. Bruckner, Differentiation of Real Functions. CRM Monograph Series (1994).
If you have an interest in derivatives then you need this updated account to take you past what was known in the 1930s as well as review of what was known and to place those ideas in an historical context. Chapter 4 gives quite a bit of information about the Dini derivatives and includes a proof of this theorem too in Theorem 4.1 p. 45.
[3] Bruckner, A. M. ; O'Malley, R. J. ; Thomson, B. S. Path derivatives: a unified view of certain generalized derivatives. Trans. Amer. Math. Soc. 283 (1984), no. 1, 97--125.
See Theorem 4.2, p. 105. This a simple generalization of Grace Young's theorem indicating a different context for thinking about such theorems..
https://www.ams.org/journals/tran/1984-283-01/S0002-9947-1984-0735410-1/S0002-9947-1984-0735410-1.pdf