## 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.

$\leftarrow$ 93-97. Limits as $x \to a$ | Main Page | 100-104. Properties of continuous functions. Bounded functions. The oscillation of a function in an interval $\rightarrow$ |