A
neighborhood of a number a is any open
interval containing a.
One common notation for a neighborhood of a is {x:
|x – a|
< δ}. Using interval
notation this would be (a – δ, a + δ).
δ = A positive real number giving the radius (half-width) of the neighborhood
x = Any point that belongs to the neighborhood
Worked Example
Problem: Describe the neighborhood of radius 0.5 centered at 3, and determine whether the points 2.8, 3.6, and 3.5 belong to it.
Step 1:Write the neighborhood using the definition with a=3 and δ=0.5.
N0.5(3)={x:∣x−3∣<0.5}=(2.5,3.5)
Step 2:Check x=2.8: compute ∣2.8−3∣=0.2. Since 0.2<0.5, the point 2.8 is inside the neighborhood.
∣2.8−3∣=0.2<0.5✓
Step 3:Check x=3.6: compute ∣3.6−3∣=0.6. Since 0.6>0.5, the point 3.6 is outside the neighborhood.
∣3.6−3∣=0.6<0.5
Step 4:Check x=3.5: compute ∣3.5−3∣=0.5. Since 0.5 is not strictly less than 0.5, the endpoint 3.5 is not inside the neighborhood. The neighborhood is an open interval, so its endpoints are excluded.
∣3.5−3∣=0.5<0.5
Answer:The neighborhood is the open interval (2.5,3.5). The point 2.8 belongs to it, while 3.6 and 3.5 do not.
Frequently Asked Questions
What is the difference between a neighborhood and a deleted neighborhood?
A neighborhood of a includes the center point a itself: (a−δ,a+δ). A deleted neighborhood removes the center point, giving (a−δ,a)∪(a,a+δ), or equivalently {x:0<∣x−a∣<δ}. Deleted neighborhoods are used when you want to examine behavior near a without considering the value at a itself, which is exactly the situation in the definition of a limit.
Why does a neighborhood have to be an open interval?
Openness ensures that every point in the neighborhood has some breathing room around it — no point sits on a boundary. This property is essential for definitions in calculus and analysis, such as limits and continuity, where you need to get arbitrarily close to a point from both sides without hitting an edge.
Neighborhood vs. Deleted Neighborhood
A neighborhood Nδ(a)=(a−δ,a+δ) contains the center point a. A deleted neighborhood Nδ∗(a)={x:0<∣x−a∣<δ} is the same set with a removed. You use a deleted neighborhood when defining limits, because the limit of f(x) as x→a depends on values near a but not at a itself.
Why It Matters
Neighborhoods are the building blocks of the formal ε-δ definitions of limits and continuity in calculus. When you read "for every ε>0 there exists a δ>0…," the statement is describing neighborhoods of specific sizes around points. Without the concept of a neighborhood, you cannot make rigorous statements about what happens "close to" a given number.
Common Mistakes
Mistake:Including the endpoints: writing [a−δ,a+δ] instead of (a−δ,a+δ).
Correction:A neighborhood is an open interval, so the boundary points a−δ and a+δ are excluded. Points at exactly distance δ from a are not in the neighborhood.
Mistake:Confusing a neighborhood with a deleted neighborhood and excluding the center point a.
Correction:A standard neighborhood includes a. Only a deleted neighborhood removes a. Be careful about which one a definition calls for — limits use deleted neighborhoods, while continuity checks use ordinary neighborhoods.
Related Terms
Open Interval — A neighborhood is a specific open interval
