Equivalent formulations of function continuity

Let $X$ be a subset of $\mathbb{R}$, let $f:X\to \mathbb{R}$ be a function, let $x_0$ be an element of $X$. Then the following four statements are logically equivalent:

1. $f$ is continuous at $x_0$.
2. [...]
3. For any real $\epsilon >0$ there exists a real $\delta >0$ such that $|f(x)-f(x_0)|<\epsilon$ for all $x\in X$ and $|x-x_0|<\delta$