# What is continuity at a point?

Let $f \left(x\right)$ be a real function of real variable defined in the domain $I \subset \mathbb{R}$.
Let ${x}_{0} \in I$ be a point of accumulation for $I$.
The function $f \left(x\right)$ is continuous in ${x}_{0}$ if:
${\lim}_{x \to {x}_{0}} f \left(x\right) = f \left({x}_{0}\right)$
that is if the limit of $f \left(x\right)$ as $x$ approaches ${x}_{0}$ equals the value of the function in ${x}_{0}$. This means that as $x$ gets closer and closer to ${x}_{0}$, then $f \left(x\right)$ gets closer and closer to $f \left({x}_{0}\right)$, so that the graph of the function has no "jumps".