after a little problem (I asked my question in the answer section, apologize, all my bad for this), I repost my question here with the same message :
"I'm new on economics stackexchange, and I've found recently a forum about differentiability of utility function (When is a utility representation differentiable ?, nov. 2015, by HRSE, link : When is a utility representation differentiable?), and I'm very interested by this question. Indeed, I've searched a lot of answers and I didn't find them : Why the fact that the space of indifferents goods $ I=\{(x,y)\in V\times V | x \sim y \} $ is a manifold of $C^{r}$ class implies that utility function is a $C^{r}$ differentiable function.
I've found many proofs about continuity, (strict) concavity/quasi-concavity of utility function (by Debreu's representation theorem mainly), but none of them show a proof of differentiability of $U(\bullet)$ function. I would like to know if we can prove it just by using the axioms of the preference relation on the set of commodities, but I can't find any solution.
I tried to show it, by using the fact that $I=\{(x,y)\in V \times V | x \succeq y\} \bigcap \{(x,y)\in V\times V | x \preceq y\}$ is a convex set of the commodities space $V$ (subset of the euclidian space $\mathbb{R}^{l},\forall l\in\mathbb{N}$) and $\succeq$ is a continuous, monotonous and convex relation on $V$ which is an open set in $ \mathbb{R}^{n}$but none of my trials has been successful.
I've made some alternative researches to prove that but I'm not convinced by them. So if you can provide any help, I would be happy ! Thank you by advance and I apologize for my bad english I'm not a native speaker.
Have a good day
EDIT 1 - Additional research and results : The set $I$ is exactly the same as the indifference set, which is continuous, convex and differentiable indifference curve (cf. $\frac{dx_{i}}{dx_{j}}=\frac{\frac{\partial U}{\partial x_{j}}}{\frac{\partial U}{\partial x_{i}}}$). But this definition implies that utility is differentiable by definition, de don't show that $U(\bullet)$ is differentiable."
Thank you all for answers, Have a nice day.
