continuous extension

If a function is not well defined on a point or set, we can extend the domain of the function.

Given a function $f:D\to \mathbb{R}$ and limit point $x_0\notin D$ ($f$ is not defined at $x_0$), such that $y:={\lim }_{x\to x_0}f(x)$ exists, we can extend $f$ to $x_0$ by defining $f(x_0):=y$. This is called a continuous extension of $f$ to $x_0$.