Continuity - Describing Functions

Mathematical Writing - Vivaldi Franco 2014

Continuity
Describing Functions

Continuity is a fundamental concept in the theory of functions. Loosely speaking, a function is continuous if it has ’no jumps’, but this naive definition is only adequate for simple situations. We begin by considering continuity at a specific point of the domain of a function. An informal—yet accurate—characterisation of continuity is to say that a function is continuous at a point if its value there can be inferred unequivocally from the values at neighbouring points. Thus neighbourhoods come into play.

For example, the value of the sign function (5.3) at the origin cannot be inferred from the surrounding environment; even if we defined, say, $$\text{ sign }(0)=1$$, the ambiguity would remain. Things can get a lot worse. Consider the real function

$$\begin{aligned} x \mapsto {\left\{ \begin{array}{ll}1 &{} {\text {if}}\, x\in \mathbb {Q}\\ 0 &{} {\text {if}}\, x\not \in \mathbb {Q}.\\ \end{array}\right. } \end{aligned}$$

(5.9)

Given that in any neighbourhood of any real number there are both rational and irrational numbers, there is no way of inferring the value of this function at any point by considering how the function behaves in the surrounding region.

To write a symbolic definition of continuity we need to quantify neighbourhoods. For this purpose, let us denote by $${\fancyscript{N}}_x$$ the set of all neighbourhoods of a point $$x$$ in $$\mathbb {R}$$. (This is an abstract set, which may be represented as a set of end-points of real intervals—see Sect. 2.3.3 and Exercise 5.12.)

We say that $$f$$ is continuous at the point $$a$$ if for every neighbourhood $$J$$ of $$f(a)$$ we can find a neighbourhood of $$a$$ whose image is contained in $$J$$. In symbols:

$$\begin{aligned} \forall J \in {\fancyscript{N}}_{f(a)},\,\,\exists I\in {\fancyscript{N}}_a, \,\,f(I)\subset J. \end{aligned}$$

(5.10)

This elaborate expression may be written in short-hand notation as

$$ \lim _{x\rightarrow a}f(x)=f(a) $$

which says that $$f(a)$$ is the limit of $$f(x)$$ as $$x$$ tends to $$a$$.

Let’s go through Definition (5.10) in slow motion. The set $${\fancyscript{N}}_a$$ consists of all open intervals containing the point $$a$$ where continuity is being tested. The set $${\fancyscript{N}}_{f(a)}$$ consists of all neighbourhoods of $$f(a)$$, the value of the function at $$a$$. An arbitrary interval $$J$$ containing $$f(a)$$ is given, typically very small. We must now find an interval $$I$$ containing $$a$$ whose image fits inside $$J$$. The choice of $$I$$ will depend on $$J$$, and if we succeed in all cases, then the function is continuous at $$a$$. We don’t need to know what the set $$f(I)$$ looks like; it suffices to know that $$f(I)$$ can be made small by making $$I$$ small.

A function that is not continuous at a point of the domain is said to be discontinuous. A function is discontinuous everywhere if it has no points of continuity, like the function (5.9).

Continuity may be defined without reference to neighbourhoods, but in this case one must supply quantitative information. The full notation is considerably more involved than (5.10):

$$ \forall \varepsilon \in \mathbb {R}^+,\,\,\exists \delta \in \mathbb {R}^+, \,\,\forall x\in \mathbb {R},\,\,|x-a|<\delta \Rightarrow |f(x)-f(a)|< \varepsilon $$

and it remains so even if we strip it of all references to $$\mathbb {R}$$:

$$\begin{aligned} \forall \varepsilon >0,\,\,\exists \delta >0,\,\,\forall x,\,\,|x-a|<\delta \, \Rightarrow \, |f(x)-f(a)|< \varepsilon . \end{aligned}$$

(5.11)

A function is continuous if it is continuous at all points of the domain. An additional quantifier is required in Definition (5.10):

$$\begin{aligned} \forall a\in A,\,\,\forall J \in {\fancyscript{N}}_{f(a)},\,\,\exists I\in {\fancyscript{N}}_a, \,\,f(I)\subset J, \end{aligned}$$

(5.12)

where $$A$$ is the domain of $$f$$.

A function $$f$$ is differentiable at $$a$$ if the limit

$$ \lim _{x\rightarrow a}\,F(x) \qquad \text{ where }\qquad F(x)=\frac{f(x)-f(a)}{x-a} \qquad x\not =a $$

exists. The function $$F$$ is called the incremental ratio of $$f$$ at $$a$$, which is not defined at $$x=a$$. However, if $$f$$ is differentiable at $$a$$, then by letting $$F(a)\mathop {=}\limits ^\mathrm{def}\lim _{x\rightarrow a}F(x)$$, the function $$F$$ becomes continuous at $$a$$. Let’s sum this up in a sentence.

A function is differentiable at a point if its incremental ratio is continuous at that point; in this case the value of the incremental ratio is the derivative of the function.

A function is said to be differentiable if it is differentiable at all points of its domain. A function that is differentiable sufficiently often (all derivatives up to a sufficiently high order exist) is said to be smooth. The expression ’sufficiently often’ is deliberately vague; its precise meaning will depend on the context.

Many elementary real functions are continuous in their respective domains: the polynomials, the trigonometric functions, and the exponential and logarithmic functions, etc. These functions are also differentiable infinitely often.

A function $$f$$ is singular at a point $$x$$ if $$f$$ is undefined at $$x$$, in which case $$x$$ is a singularity (or a singular point) of the function. So the function $$x\mapsto 1/x$$ is singular at the origin. A function that is not singular is said to be well-behaved or regular, the latter term used mostly for functions of a complex variable. We write:

A rational function has finitely many singularities: the roots of the polynomial at the denominator. This function is regular everywhere else.

The tangent, secant, and co-secant have infinitely many singularities, evenly spaced along the real line.

In analysis the term ’singular’ is also used for well-defined functions that exhibit certain esoteric pathologies.