Neighbourhood of a point

To speak of “closeness” to a point in a precise way we introduce the neighbourhood.

Definition — Neighbourhood

Given $x_0\in\mathbb{R}$ and $\delta>0$, the (open) neighbourhood centred at $x_0$ with radius $\delta$ is the interval $I_\delta(x_0) = (x_0-\delta,\ x_0+\delta) = \{x\in\mathbb{R} : |x-x_0|<\delta\}.$ The punctured neighbourhood is $\dot I_\delta(x_0) = I_\delta(x_0)\setminus\{x_0\}$: the same interval with its centre removed.

The punctured neighbourhood is needed because, when we study a limit at $x_0$, we are interested in the behaviour of the function around $x_0$ but not at the point itself.

Links

Topics: Limits
Concepts: Neighbourhood · Absolute value