<p>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 :</p>
<p>&quot;I'm new on economics stackexchange, and I've found recently a forum about differentiability of utility function (<em>When is a utility representation differentiable ?</em>, nov. 2015, by HRSE, <strong>link</strong> : <a href="https://economics.stackexchange.com/questions/9158/when-is-a-utility-representation-differentiable/57651?noredirect=1#comment94663_57651">When is a utility representation differentiable?</a>), 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 <span class="math-container">$ I=\{(x,y)\in V\times V | x \sim y \} $</span> is a manifold of <span class="math-container">$C^{r}$</span> class implies that utility function is a <span class="math-container">$C^{r}$</span> differentiable function.</p>
<p>I've found many proofs about continuity, (strict) concavity/quasi-concavity of utility function (by Debreu's representation theorem mainly), but none of them show a proof of differentiability of <span class="math-container">$U(\bullet)$</span> function. I would like to know if we can prove it just by using the axioms of the preference relation on the set of commodities, but I can't find any solution.</p>
<p>I tried to show it, by using the fact that <span class="math-container">$I=\{(x,y)\in V \times V | x \succeq y\} \bigcap \{(x,y)\in V\times V | x \preceq y\}$</span> is a convex set of the commodities space <span class="math-container">$V$</span> (subset of the euclidian space <span class="math-container">$\mathbb{R}^{l},\forall l\in\mathbb{N}$</span>) and <span class="math-container">$\succeq$</span> is a continuous, monotonous and convex relation on <span class="math-container">$V$</span> which is an open set in <span class="math-container">$ \mathbb{R}^{n}$</span>but none of my trials has been successful.</p>
<p>I've made some alternative researches to prove that but I'm not convinced by them. So if you can provide any help, I would be happy ! Thank you by advance and I apologize for my bad english I'm not a native speaker.</p>
<p>Have a good day</p>
<p>EDIT 1 - Additional research and results : The set <span class="math-container">$I$</span> is exactly the same as the indifference set, which is continuous, convex and differentiable indifference curve (cf. <span class="math-container">$\frac{dx_{i}}{dx_{j}}=\frac{\frac{\partial U}{\partial x_{j}}}{\frac{\partial U}{\partial x_{i}}}$</span>). But this definition implies that utility is differentiable by definition, de don't show that <span class="math-container">$U(\bullet)$</span> is differentiable.&quot;</p>
<p>Thank you all for answers,
Have a nice day.</p>

CC BY-SA 4.0
2024-02-28T18:07:45.083Z
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.
Michael Greinecker
CC BY-SA 4.0
2024-03-03T13:39:08.277Z
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.
Economos
<microeconomics><mathematical-economics><utility>
Proof that utility function is differentiable
Economos
# Proof that utility function is differentiable

(Asked by: username_0 on 2024-02-28)

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?](https://economics.stackexchange.com/questions/9158/when-is-a-utility-representation-differentiable/57651?noredirect=1#comment94663_57651)), 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.

**Tags:** microeconomics, mathematical-economics, utility

> **Comments:**
> **username_1**: 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.  
> **username_0**: 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.  

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