Neighborhood of a Point: Definition, Examples and Limits

On the real number line, a δ-neighborhood of a point a is the open interval (a − δ, a + δ), where δ > 0. It contains every number whose distance from a is less than δ. It includes a itself, but neither endpoint. “Neighbourhood” is the British spelling of the same term.

A deleted neighborhood, also called a punctured neighborhood, removes the center a. That distinction matters in limits: we look at nearby inputs without requiring the input to equal a.

What does a neighborhood mean in calculus?

The positive radius δ (delta) sets the allowed distance from the center. These three descriptions mean the same thing:

|x − a| < δ
a − δ < x < a + δ
x belongs to (a − δ, a + δ)

The vertical bars mean absolute value, so distance is measured on either side of a. The strict inequality excludes both endpoints. A radius may be large or small; the definition does not require one particular size.

Example: a neighborhood of 2

Choose a = 2 and δ = 0.5. The neighborhood is (1.5, 2.5). Its total width is 1, twice the radius.

For x = 2.3, |2.3 − 2| = 0.3 < 0.5, so 2.3 is inside. The center 2 is inside because its distance is 0. The endpoints 1.5 and 2.5 are outside because their distance equals 0.5 rather than being less than it.

Ordinary versus deleted neighborhoods

An ordinary neighborhood includes the center. A deleted neighborhood excludes it:

0 < |x − a| < δ
(a − δ, a) ∪ (a, a + δ)

The symbol ∪ joins the two open intervals. In our example, the deleted neighborhood is (1.5, 2) ∪ (2, 2.5). Now 2.3 still belongs, but 2 does not. Neither endpoint belongs in either version.

How do neighborhoods describe a limit?

For a function defined near a, the statement lim as x → a of f(x) = L means that every positive output tolerance ε (epsilon) can be met by choosing a positive input radius δ. Whenever 0 < |x − a| < δ, we then have |f(x) − L| < ε.

Delta controls inputs near a; epsilon controls outputs near L. Delta can depend on epsilon. The condition excludes x = a, so f(a) can differ from L or be undefined while the limit still exists.

A short worked limit example

Let f(x) = 3x − 2 and a = 4. The proposed limit is 10. Since |f(x) − 10| = |3x − 12| = 3|x − 4|, choose δ = ε/3. Then 0 < |x − 4| < δ implies |f(x) − 10| < 3δ = ε.

For instance, ε = 0.6 allows δ = 0.2: inputs in the deleted interval around 4 give outputs within 0.6 of 10. This works for every ε > 0, which is what the definition requires.

For the formal statement and further explanation, see OpenStax’s precise definition of a limit. This page uses symmetric intervals on the real line; more general neighborhoods in topology require a broader definition.

Check your understanding

If a = −1 and δ = 2, what is the ordinary neighborhood? Does it contain 0, −1 or 1?

Continue with the free Calculus lessons and practice hub to choose a related limits topic. Use the definition above to separate “near the point” from “at the point” before moving on.

Existing video explanation

ε-neighborhoods and neighborhoods in higher dimensions

The ε-neighborhood of a is written N_ε(a) or B(a, ε). On the number line it is the open interval (a − ε, a + ε), all points closer to a than ε. Some books use δ instead of ε; the idea is the same.

In general, a set N is a neighborhood of p if N contains some ε-neighborhood of p. So [0, 2] is a neighborhood of 1, because it contains (0.5, 1.5). It is not a neighborhood of 0, since every interval around 0 includes negative numbers outside [0, 2].

Neighborhood of a set

A set N is a neighborhood of a set S if N contains an open set that contains all of S. Put simply, N must have a little room around every point of S.

The ε-neighborhood of a set S is the set of all points within distance ε of some point of S. For S = [0, 1] and ε = 0.1, it is (−0.1, 1.1). That interval is a neighborhood of [0, 1]. The set [0, 1] itself is not a neighborhood of [0, 1], because it gives no room around the endpoints 0 and 1.

For limits, you mostly need neighborhoods of single points. Neighborhoods of sets appear later in analysis and topology, for example when proving that a continuous function stays close to its values on a whole interval.

Frequently Asked Questions

What is a neighborhood of a point?

A neighborhood of a point a is a set that contains an open interval around a. On the real line, the δ-neighborhood of a is (a − δ, a + δ), the numbers whose distance from a is less than δ, where δ > 0. For a = −1 and δ = 0.25, the neighborhood is (−1.25, −0.75). It contains a but not the endpoints.

What is a deleted neighbourhood of a point?

A deleted, or punctured, neighbourhood is an ordinary neighbourhood with the center point removed. On the real line it is {x : 0 < |x − a| < δ}, which equals (a − δ, a) ∪ (a, a + δ). For a = 5 and δ = 0.2, it is (4.8, 5) ∪ (5, 5.2). Limits use deleted neighbourhoods because the value of a function at a itself does not affect the limit.

What is an epsilon neighborhood?

An epsilon neighborhood of a point a is the set of all points within a distance ε of a, where ε > 0. On the real line it is the open interval (a − ε, a + ε), written {x : |x − a| < ε}. For a = 0 and ε = 0.01 it is (−0.01, 0.01). In the definition of a limit, ε usually sets how close the outputs must be to L.

What are some examples of a neighbourhood of a point?

The interval (0, 3) is a neighbourhood of 1, because it contains (0.5, 1.5), an open interval centered at 1. The whole real line is a neighbourhood of every point. But [1, 3] is not a neighbourhood of 1, since no open interval around 1 fits inside it: any interval (1 − δ, 1 + δ) includes numbers less than 1.

What is a neighbourhood of a point in a metric space?

In a metric space with distance d, the open ball B(p, r) = {x : d(x, p) < r} is the basic neighbourhood of a point p, for any radius r > 0. A neighbourhood of p is any set containing such a ball. In the plane with ordinary distance, the open disk of radius 1 around (2, 3) is a neighbourhood of (2, 3), and on the real line the ball is just an open interval.

What is a neighbourhood of a point in topology?

In a topological space, a set N is a neighbourhood of a point p if N contains an open set U with p in U. Under this general definition a neighbourhood does not have to be open itself. For example, on the real line the closed interval [−1, 1] is a neighbourhood of 0 because it contains the open interval (−0.5, 0.5). Some textbooks require neighbourhoods to be open, so check yours.