# Proof that utility function is differentiable

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.

• The relevant answer, which happens to be mine, has a reference to Mas-Colell's book The Theory of General Economic Equilibrium: A Differentiable Approach". You can find the proof there. It is quite long, dense, and involved. This is not a simple exercise. Feb 28 at 18:07
• Hello, thank you for your answer I'll try to find it if it is the only proof we can have here for this question. Mar 3 at 13:39