Neighbourhoods - Neighbourhoods and Sets - Describing Functions

Mathematical Writing - Vivaldi Franco 2014

Neighbourhoods - Neighbourhoods and Sets
Describing Functions

A neighbourhood of a point $$x\in \mathbb {R}$$ is any open interval containing $$x$$. Although this definition makes no reference to the size of the interval, a neighbourhood of $$x$$ contains all points sufficiently close to $$x$$. Thus the neighbourhood concept characterises ’proximity’ in a concise manner that does not require quantitative information. A skillful use of this term leads to terse and incisive statements. For instance, the sentence

The function $$f$$ is bounded in a neighbourhood of $$x_0$$.

means that there is an open interval containing $$x_0$$ whose image under $$f$$ is a bounded set. If we write this statement in symbols

$$ \exists a,b\in \mathbb {R},\,\,\exists c\in \mathbb {R}^+,\,\,x_0\in (a,b)\wedge (\forall x \in (a,b),\,\,|f(x)|<c) $$

we realise just how much information is packed into it. The following variant of the sentence above:

The function $$f$$ is bounded in a sufficiently small neighbourhood of $$x_0$$.

says exactly the same thing, but more eloquently; the reader is warned that the required neighbourhood may be very small.

A property of a function—boundedness in this case—which holds in a neighbourhood of some point of the domain of the function, but not necessarily in the whole domain, is said to be local. So a function may be locally increasing, locally injective, etc.

The function $$f$$ has a maximum at $$x$$ if the value of $$f$$ at $$x$$ is greater than the value at all other points, namely if

$$\begin{aligned} \forall y\in \mathbb {R}{\backslash }\{x\},\,\,f(x)>f(y). \end{aligned}$$

(5.8)

The function $$f$$ has a local maximum at $$x$$ if (5.8) holds in some neighbourhood of $$x$$. The concept of minimum and local minimum are defined similarly. Thus the exponential has no maximum or minimum, the hyperbolic cosine has a minimum but no maximum, and the function $$x\mapsto \arctan (x)+\sin (x)$$ has infinitely many local maxima and minima, but no maximum or minimum.

By a neighbourhood of infinity we mean a ray $$\{x\in \mathbb {R}:x>a\}$$, where $$a$$ is a real number. A neighbourhood of $$-\infty $$ is defined similarly, and both points at infinity $$\pm \infty $$ are handled at once with the construct $$\{x\in \mathbb {R}:|x|>a\}$$. The points $$\pm \infty $$ are not numbers but they have neighbourhoods, which make them more tangible. So the sentence

The function $$f$$ is constant in a neighbourhood of infinity.

means that $$f$$ is constant for all sufficiently large values of the argument. It may be written symbolically as

$$ \exists a\in \mathbb {R},\,\,\forall x\in \mathbb {R}^+,\,\,f(a+x)=f(a) $$

or as

$$ \exists a\in \mathbb {R},\,\,\forall x>a,\,\,f(x)=f(a). $$

5.4.1 Neighbourhoods and Sets

Neighbourhoods are instrumental to the description of sets of numbers. This is an appealing part of the mathematical dictionary, due to the vivid mental pictures we associate with a geometric language. The case for expanding our dictionary is easily made:

$$ X_1=\{1/n\,:\,n\in \mathbb {N}\}\qquad X_2=\mathbb {Q}\cap (0,1). $$

How can we describe such sets?

Let $$X\subset \mathbb {R}$$. A point $$x\in X$$ is isolated if there is a neighbourhood of $$x$$ that contains no other point of $$X$$. The set $$\mathbb {Z}$$ consists entirely of isolated points, and so does the set $$X_1$$ above. By contrast, $$X_2$$ has no isolated points.

A point $$x$$ is an interior point of a set $$X$$ if $$X$$ contains a neighbourhood of $$x$$. A point $$x$$ is a boundary point of a set $$X$$ if every neighbourhood of $$x$$ contains points of $$X$$ as well as points of the complement of $$X$$. An isolated point is necessarily a boundary point, but not all boundary points are isolated. The boundary points of an interval are its end-points; all other points are interior points and there are no isolated points. Neither $$X_1$$ nor $$X_2$$ have interior points; the origin is a limit point of $$X_1$$.

A set is closed if it contains all its boundary points, and is open if all its points are interior points. For intervals, these concepts agree with those given in Sect. 2.1.3. The sets $$X_1$$ and $$X_2$$ are neither open nor closed. The closure of a set $$X$$, denoted by $$\overline{X}$$, is the union of $$X$$ and the boundary points of $$X$$. We see that $$\overline{X}_1=X_1\cup \{0\}$$ and $$\overline{X}_2=[0,1]$$.