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.

approximate differentiabilityof a function $f:E\subset\mathbb R^n\to\mathbb R^m$? (It seems that if we take corresponding norms on both terms of the fraction in the above definition of approximate differentiability, then appriximate differentiability can be defined). Also, is it available in the literature? $\endgroup$ – Biswa Ranjan Datt Feb 19 '20 at 12:11