I am wondering if this statement below is true?
Let $f\colon D\subset \mathbb{R}\to\mathbb{R},$ and $x_0\in\mathbb{R}$ be a limit point of $D,$ and $A\in\mathbb{R}.$ Then $\lim_{x\to x_0} f(x)=A,$ if and only if for every neighborhood $\mathbb{B}(A,\epsilon)$ of $A,$ its preimage $f^{-1}(\mathbb{B}(A,\epsilon))$ is a deleted neighborhood of $x_0.$
Here $\mathbb{B}(A,\epsilon)=\{y\in\mathbb{R}\mid |y-A|<\epsilon\}.$ And we call $U$ is a deleted neighborhood of $x_0,$ if there exists an open set $O$ such that $(O\backslash\{x_0\})\subset U.$