Real valued function continuous at a point

Let $f: I \to \mathbb{R}$ be a real valued function on an interval.

Assuming that $f$ is defined near $a \in \mathbb{R}$, we say that $f(x)$ is continuous at $x=a$ or continuous at $a$ if additionally,

1. $f(x)$ is defined at $x = a$.
2. The limit $\lim\limits_{x \to a} f(x)$ exists, and
3. $f(a) = \lim\limits_{x \to a} f(x)$.