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Define $f: [a,b]\to \Bbb R$, we can talk about the derivative of $f$ at the boundary points $a$ and $b$.

I am interested in the case when $f$ is a function of several variables.

Define $f:E\to \Bbb R^m$ with $E\subset\Bbb R^n$. One can talk about the derivative of $f$ only if $E$ is open. I have no idea why this is the case.

The answer here: https://math.stackexchange.com/a/504567/135775, says that if $E$ is not open the the Jacobian of $f$ need not be unique.

Can someone give an explanation of why one cannot define the derivative if $E$ is closed or maybe an example where the Jacobian is not unique.

Naive
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    If $E$ is closed, it's easy to construct functions that don't have derivatives on the boundary - but it's no different to $\mathbb{R}$. e.g. $f:[0,1]\rightarrow \mathbb{R}, f(x) = \sqrt{x}$. I spent about an hour trying to create an example in $\mathbb{R}^2$ with a non-unique jacobian but to no avail. – Harambe Nov 09 '17 at 21:01

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