Sufficient conditions on E to ensure equality of the Frechet derivatives in E of two functions f,g that agree in E.

Question

Let $E$ be a subset of $[0,1]^n$ which is Lebesgue measurable with positive measure. Let $f,g$ be two Lipschitz functions on $[0,1]^n$ that agree in $E$: $f(x)=g(x)$ for all $x\in E$.

I am wondering about sufficient conditions on $E$ that would grant equality of the Frechet derivatives in $E$ (up to a zero measure set).

Here is some thoughts:

1. The Frechet derivatives of $f,g$ exist almost everywhere by Rademacher theorem. We are looking for conditions that grant equality of the Frechet derivatives, or equivalently of all partial derivatives where the Frechet derivatives exist.
2. If $E$ is open, then the derivatives must be equal in $E$ whenever one derivative (whenever either the derivative of $g$ or that of $f$ exists).
3. If $E$ has a dense neighborhood around every $x\in E$, then the derivatives also agree. For instance if $E$ contains all rationals then the derivatives must be equal in $E$ wherever they exist.
4. I am especially worried about nowhere dense sets of positive Lebesgue measure, cf. https://en.wikipedia.org/wiki/Nowhere_dense_set#Nowhere_dense_sets_with_positive_measure
5. One can possibly use the fact that $f,g$ are absolutely continuous on almost every lines parallel to the coordinate axis to reduce the problem to the one dimensional case.

So I am essentially looking for weaker conditions than $E$ being open. It is fine to remove subsets of $E$ of measure $0$.