Define continuity of a function at a point.

Continuity at a point:

A function $f(x)$ is said to be continuous at a point $x=a$ of its domain, iff $\lim _{x \rightarrow a} f(x)=f(a)$.

Thus, $f(x)$ is continuous at $x=a$.

$\Leftrightarrow \lim _{x \rightarrow a} f(x)=f(a) \Leftrightarrow \lim _{x \rightarrow a^{-}} f(x)=\lim _{x \rightarrow a^{+}} f(x)=f(a)$