The classical definition of an approximately differentiable function is as follows:

Definition.Let $f:E\to\mathbb{R}$ be a measurable function defined on a measurable set $E\subset\mathbb{R}^n$. We say that $f$ is approximately differentiable at $x\in E$ if there is a linear function $L:\mathbb{R}^n\to\mathbb{R}$ such that for any $\varepsilon>0$ the set $$ \left\{ y\in E:\, \frac{|f(y)-f(x)-L(y-x)|}{|y-x|} <\varepsilon \right\} \quad \text{has $x$ as a density point.} $$

It turns out that this definition is equivalent to the one described in the following result.

Theorem.A measurable function $f:E\to\mathbb{R}$ defined in a measurable set $E\subset\mathbb{R}^n$ is approximately differentiable at $x\in E$ if and only if there is a measurable set $E_x\subset E$ and a linear function $L:\mathbb{R}^n\to\mathbb{R}$ such that $x$ is a density point of $E_x$ and $$ \lim_{E_x\ni y\to x} \frac{|f(y)-f(x)-L(y-x)|}{|y-x|} = 0. $$

**Do you know any reference for the proof of this result?** Since we could not find a good reference, we proved it in the appendix of the paper listed below, but I am pretty sure it can be found somewhere else.

**P. Goldstein, P. Hajłasz**, A measure and orientation preserving homeomorphism with approximate Jacobian equal −1 almost everywhere.
*Arch. Ration. Mech. Anal.* 225 (2017), 65–88.

In my opinion it is a folklore result that most of the people working with approximately differentiable functions know. The characterization given in the theorem is much easier to use than the original definition.