# On Choquet's characterization of antiderivatives

According to Wikipedia, G. Choquet has characterized all real functions which, up to a bicontinuous change of variable, have antiderivative as all the Baire 1 functions which send intervals onto connected sets. I am quite confused by this change of variable. It seems quite clear to me that if a function satisfies the two last conditions, the same holds for all its bicontinuous changes of variable. Can I see an example of a function which has antiderivative, but such that a bicontinuous change of variable of it has no antiderivative? I would expect that such a change of variable is not differentiable, but I have a hard time constructing an example. I have searched G. Choquet's work on internet, but I could not find it either. Is there any reasonably good book discussing the matter in sufficient detail?

(This question was posted with still no reactions on Stack Exchange 4 days ago.)

• Where did you find this statement? – LSpice Apr 25 at 12:26
• – Hair80 Apr 25 at 13:06

## 1 Answer

A possible place to look for examples:

Bruckner, Andrew M., Differentiation of real functions, CRM Monograph Series. 5. Providence, RI: American Mathematical Society (AMS). xi, 195 p. (1994). ZBL0796.26004.

I do not remember enough to be more specific.

• As only a possible reference, this seems like a comment, not an answer. – LSpice Apr 25 at 12:24
• In Bruckner's book mentioned above, the following example is elaborated on page 14. Let $F(x)= x^2 \sin(1/x^2)$, $F(0)=0$, and $f=F'$. Then $f^2$ is not a derivative, but it is Baire-1 and has the intermediate value property (since $f$ has these properties). So one cannot dispense with the change of variable requirement in Choquet's theorem. Bruckner also refers to two papers by Maximoff from the 1940s containing the same result as the one that Choquet proved in 1947. – Dirk Werner Apr 30 at 17:20