I found the definition of approximate continuity stated as follows:

A function $f:\mathbb R \to \mathbb R$ is **approximately continuous** at $x_0$ iff there exists a set $A\in \mathcal L$ such that $x_0\in \Phi(A)$ and $$\lim\limits_{x\to x_0,\ x\in A}f(x)=f(x_0)$$

where, $\mathcal L$ is the set of all Lebesgue measurable subsets in $\mathbb R$, and $\Phi(A)$ is the set of all density points of $A\subset \mathbb R$.

## Question1: Can we write the above definition in $\epsilon$-$\delta$ form as follows?

A function $f:\mathbb R \to \mathbb R$ is **approximately continuous** at $x_0$ if and only if for each $\epsilon>0$ there exist $\delta>0$ and $A\in \mathcal L$ with $x_0\in \Phi(A)$ such that $$|f(x)-f(x_0)|<\epsilon\quad \text{whenever}\quad x\in (x_0-\delta, x_0+\delta)\cap A$$

## Question2: Also, can we write the definition of "Approximate continuity" as follow?

A function $f:\mathbb R \to \mathbb R$ is **approximately continuous** at $x_0$ if and only if for each $\epsilon>0$ the set $\{y\in \mathbb R: |f(y)-f(x_0)|<\epsilon\}$ has $x_0$ as a density point.