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. For every sequence $(a_n{)}_{n=0}^{\mathrm{\infty }}$ consisting of elements of $X$ which converges to $x_0$, the sequence $(f(a_n){)}_{n=0}^{\mathrm{\infty }}$ converges to $f(x_0)$.
3. [...]