98-99. Continuous functions of a real variable

98. Continuous functions of a real variable.

The reader has no doubt some idea as to what is meant by a continuous curve. Thus he would call the curve $C$ in Fig. 29 continuous, the curve $C’$ generally continuous but discontinuous for $x = \xi’$ and $x = \xi”$.

Either of these curves may be regarded as the graph of a function $\phi(x)$. It is natural to call a function continuous if its graph is a continuous curve, and otherwise discontinuous. Let us take this as a provisional definition and try to distinguish more precisely some of the properties which are involved in it.

In the first place it is evident that the property of the function $y = \phi(x)$ of which $C$ is the graph may be analysed into some property possessed by the curve at each of its points. To be able to define continuity for all values of $x$ we must first define continuity for any particular value of $x$. Let us therefore fix on some particular value of $x$, say the value $x = \xi$ corresponding to the point $P$ of the graph. What are the characteristic properties of $\phi(x)$ associated with this value of $x$?

In the first place $\phi(x)$ is defined for $x = \xi$. This is obviously essential. If $\phi(\xi)$ were not defined there would be a point missing from the curve.

Secondly $\phi(x)$ is defined for all values of $x$ near $x = \xi$; we can find an interval, including $x = \xi$ in its interior, for all points of which $\phi(x)$ is defined.

Thirdly if $x$ approaches the value $\xi$ from either side then $\phi(x)$ approaches the limit $\phi(\xi)$.

The properties thus defined are far from exhausting those which are possessed by the curve as pictured by the eye of common sense. This picture of a curve is a generalisation from particular curves such as straight lines and circles. But they are the simplest and most fundamental properties: and the graph of any function which has these properties would, so far as drawing it is practically possible, satisfy our geometrical feeling of what a continuous curve should be. We therefore select these properties as embodying the mathematical notion of continuity. We are thus led to the following

Definition. The function $\phi(x)$ is said to be continuous for $x = \xi$ if it tends to a limit as $x$ tends to $\xi$ from either side, and each of these limits is equal to $\phi(\xi)$.

We can now define continuity throughout an interval. The function $\phi(x)$ is said to be continuous throughout a certain interval of values of $x$ if it is continuous for all values of $x$ in that interval. It is said to be continuous everywhere if it is continuous for every value of $x$. Thus $[x]$ is continuous in the interval ${[\delta, 1 – \delta]}$, where $\delta$ is any positive number less than $\frac{1}{2}$; and $1$ and $x$ are continuous everywhere.

If we recur to the definitions of a limit we see that our definition is equivalent to ‘$\phi(x)$ is continuous for $x = \xi$ if, given $\epsilon$, we can choose $\delta(\epsilon)$ so that $|\phi(x) – \phi(\xi)| < \epsilon$ if $0 \leq |x – \xi| \leq \delta(\epsilon)$’.

We have often to consider functions defined only in an interval ${[a, b]}$. In this case it is convenient to make a slight and obvious change in our definition of continuity in so far as it concerns the particular points $a$ and $b$. We shall then say that $\phi(x)$ is continuous for $x = a$ if $\phi(a + 0)$ exists and is equal to $\phi(a)$, and for $x = b$ if $\phi(b – 0)$ exists and is equal to $\phi(b)$.

99.

The definition of continuity given in the last section may be illustrated geometrically as follows. Draw the two horizontal lines $y = \phi(\xi) – \epsilon$ and $y = \phi(\xi) + \epsilon$. Then $|\phi(x) – \phi(\xi)| < \epsilon$ expresses the fact that the point on the curve corresponding to $x$ lies between these two lines. Similarly $|x – \xi| \leq \delta$ expresses the fact that $x$ lies in the interval ${[\xi – \delta, \xi + \delta]}$. Thus our definition asserts that if we draw two such horizontal lines, no matter how close together, we can always cut off a vertical strip of the plane by two vertical lines in such a way that all that part of the curve which is contained in the strip lies between the two horizontal lines. This is evidently true of the curve $C$ (Fig. 29), whatever value $\xi$ may have.

We shall now discuss the continuity of some special types of functions. Some of the results which follow were (as we pointed out at the time) tacitly assumed in Ch. II.

Example XXXVII

1. The sum or product of two functions continuous at a point is continuous at that point. The quotient is also continuous unless the denominator vanishes at the point. [This follows at once from Ex. XXXV. 1.]

2. Any polynomial is continuous for all values of $x$. Any rational fraction is continuous except for values of $x$ for which the denominator vanishes. [This follows from Exs. XXXV. 6, 7.]

3. $\sqrt{x}$ is continuous for all positive values of $x$ (Ex. XXXV. 8). It is not defined when $x < 0$, but is continuous for $x = 0$ in virtue of the remark made at the end of § 98. The same is true of $x^{m/n}$, where $m$ and $n$ are any positive integers of which $n$ is even.

4. The function $x^{m/n}$, where $n$ is odd, is continuous for all values of $x$.

5. $1/x$ is not continuous for $x = 0$. It has no value for $x = 0$, nor does it tend to a limit as $x \to 0$. In fact $1/x \to +\infty$ or $1/x \to -\infty$ according as $x \to 0$ by positive or negative values.

6. Discuss the continuity of $x^{-m/n}$, where $m$ and $n$ are positive integers, for $x = 0$.

7. The standard rational function $R(x) = P(x)/Q(x)$ is discontinuous for $x = a$, where $a$ is any root of $Q(x) = 0$. Thus $(x^{2} + 1)/(x^{2} – 3x + 2)$ is discontinuous for $x = 1$. It will be noticed that in the case of rational functions a discontinuity is always associated with (a) a failure of the definition for a particular value of $x$ and (b) a tending of the function to $+\infty$ or $-\infty$ as $x$ approaches this value from either side. Such a particular kind of point of discontinuity is usually described as an infinity of the function. An ‘infinity’ is the kind of discontinuity of most common occurrence in ordinary work.

8. Discuss the continuity of \[\sqrt{(x – a)(b – x)},\quad \sqrt[3]{(x – a)(b – x)},\quad \sqrt{(x – a)/(b – x)},\quad \sqrt[3]{(x – a)/(b – x)}.\]

9. $\sin x$ and $\cos x$ are continuous for all values of $x$.

[We have \[\sin(x + h) – \sin x = 2\sin \tfrac{1}{2}h \cos(x + \tfrac{1}{2}h),\] which is numerically less than the numerical value of $h$.]

10. For what values of $x$ are $\tan x$, $\cot x$, $\sec x$, and $\csc x$ continuous or discontinuous?

11. If $f(y)$ is continuous for $y = \eta$, and $\phi(x)$ is a continuous function of $x$ which is equal to $\eta$ when $x = \xi$, then $f\{\phi(x)\}$ is continuous for $x = \xi$.

12. If $\phi(x)$ is continuous for any particular value of $x$, then any polynomial in $\phi(x)$, such as $a\{\phi(x)\}^{m} + \dots$, is so too.

13. Discuss the continuity of \[1/(a\cos^{2} x + b\sin^{2} x),\quad \sqrt{2 + \cos x},\quad \sqrt{1 + \sin x},\quad 1/\sqrt{1 + \sin x}.\]

14. $\sin(1/x)$, $x\sin(1/x)$, and $x^{2}\sin(1/x)$ are continuous except for $x = 0$.

15. The function which is equal to $x\sin(1/x)$ except when $x = 0$, and to zero when $x = 0$, is continuous for all values of $x$.

16. $[x]$ and $x – [x]$ are discontinuous for all integral values of $x$.

17. For what (if any) values of $x$ are the following functions discontinuous: $[x^{2}]$, $[\sqrt{x}\,]$, $\sqrt{x – [x]}$, $[x] + \sqrt{x – [x]}$, $[2x]$, $[x] + [-x]$?

18. Classification of discontinuities. Some of the preceding examples suggest a classification of different types of discontinuity.

(1) Suppose that $\phi(x)$ tends to a limit as $x \to a$ either by values less than or by values greater than $a$. Denote these limits, as in § 95, by $\phi(a – 0)$ and $\phi(a + 0)$ respectively. Then, for continuity, it is necessary and sufficient that $\phi(x)$ should be defined for $x = a$, and that $\phi(a – 0) = \phi(a) = \phi(a + 0)$. Discontinuity may arise in a variety of ways.

($\alpha$) $\phi(a – 0)$ may be equal to $\phi(a + 0)$, but $\phi(a)$ may not be defined, or may differ from $\phi(a – 0)$ and $\phi(a + 0)$. Thus if $\phi(x) = x \sin(1/x)$ and $a = 0$, $\phi(0 – 0) = \phi(0 + 0) = 0$, but $\phi(x)$ is not defined for $x = 0$. Or if $\phi(x) = [1 – x^{2}]$ and $a = 0$, $\phi(0 – 0) = \phi(0 + 0) = 0$, but $\phi(0) = 1$.

($\beta$) $\phi(a – 0)$ and $\phi(a + 0)$ may be unequal. In this case $\phi(a)$ may be equal to one or to neither, or be undefined. The first case is illustrated by $\phi(x) = [x]$, for which $\phi(0 – 0) = -1$, $\phi(0 + 0) = \phi(0) = 0$; the second by $\phi(x) = [x] – [-x]$, for which $\phi(0 – 0) = -1$, $\phi(0 + 0) = 1$, $\phi(0) = 0$; and the third by $\phi(x) = [x] + x \sin(1/x)$, for which $\phi(0 – 0)= -1$, $\phi(0 + 0) = 0$, and $\phi(0)$ is undefined.

In any of these cases we say that $\phi(x)$ has a at $x = a$. And to these cases we may add those in which $\phi(x)$ is defined only on one side of $x = a$, and $\phi(a – 0)$ or $\phi(a + 0)$, as the case may be, exists, but $\phi(x)$ is either not defined when $x = a$ or has when $x = a$ a value different from $\phi(a – 0)$ or $\phi(a + 0)$.

It is plain from § 95 that a function which increases or decreases steadily in the neighbourhood of $x = a$ can have at most a simple discontinuity for $x = a$.

(2) It may be the case that only one (or neither) of $\phi(a – 0)$ and $\phi(a + 0)$ exists, but that, supposing for example $\phi(a + 0)$ not to exist, $\phi(x) \to +\infty$ or $\phi(x) \to -\infty$ as $x \to a+0$, so that $\phi(x)$ tends to a limit or to $+\infty$ or to $-\infty$ as $x$ approaches $a$ from either side. Such is the case, for instance, if $\phi(x) = 1/x$ or $\phi(x) = 1/x^{2}$, and $a = 0$. In such cases we say (cf. Ex. 7) that $x = a$ is an of $\phi(x)$. And again we may add to these cases those in which $\phi(x) \to +\infty$ or $\phi(x) \to -\infty$ as $x \to a$ from one side, but $\phi(x)$ is not defined at all on the other side of $x = a$.

(3) Any point of discontinuity which is not a point of simple discontinuity nor an infinity is called a point of . Such is the point $x = 0$ for the functions $\sin(1/x)$, $(1/x)\sin(1/x)$.

19. What is the nature of the discontinuities at $x = 0$ of the functions $(\sin x)/x$, $[x] + [-x]$, $\csc x$, $\sqrt{1/x}$, $\sqrt[3]{1/x}$, $\csc(1/x)$, $\sin(1/x)/\sin(1/x)$?

20. The function which is equal to $1$ when $x$ is rational and to $0$ when $x$ is irrational (Ch. II, Ex. XVI. 10) is discontinuous for all values of $x$. So too is any function which is defined only for rational or for irrational values of $x$.

21. The function which is equal to $x$ when $x$ is irrational and to $\sqrt{(1 + p^{2})/(1 + q^{2})}$ when $x$ is a rational fraction $p/q$ (Ch. II, Ex. XVI. 11) is discontinuous for all negative and for positive rational values of $x$, but continuous for positive irrational values.

22. For what points are the functions considered in Ch. IV, Exs. XXXI discontinuous, and what is the nature of their discontinuities? [Consider, , the function $y = \lim x^{n}$ (Ex. 5). Here $y$ is only defined when $-1 < x \leq 1$: it is equal to $0$ when $-1 < x < 1$ and to $1$ when $x = 1$. The points $x = 1$ and $x = -1$ are points of simple discontinuity.]

$\leftarrow$ 93-97. Limits as $x \to a$

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100-104. Properties of continuous functions. Bounded functions. The oscillation of a function in an interval $\rightarrow$