Epsilon Neighbourhood (Scalar)

analysis

Definition

Epsilon Neighbourhood (Scalar)

The epsilon neighbourhood of a point $a\in \mathbb{R}$, denoted by $U_{\epsilon }(a)$, is the set of all points whose distance from $a$ is strictly less than $\epsilon$:

$$U_{\epsilon }(a)=(a-\epsilon ,a+\epsilon )=\{x\in \mathbb{R}\mid |x-a|<\epsilon \}$$

Geometrically, this corresponds to an open interval centred at $a$ with radius $\epsilon$.

Contextual Usage

• Convergence: A sequence $(a_n)$ converges to $L$ if for every $\epsilon >0$, the neighbourhood $U_{\epsilon }(L)$ contains almost all terms of the sequence.
• Continuity: A function $f$ is continuous at $x_0$ if for every epsilon neighbourhood of $f(x_0)$, there exists a delta neighbourhood of $x_0$ such that $f(U_{\delta }(x_0))\subseteq U_{\epsilon }(f(x_0))$.
• Topological Definition: A set is open if every point in the set has an epsilon neighbourhood contained entirely within the set.