Let us think of some examples of functions presented to us in nature, so as to get into our heads the real bearing of continuity and discontinuity. Consider a train in its journey along a railway line, say from Euston Station, the terminus in London of the London and North-Western Railway. Along the line in order lie the stations of Bletchley and Rugby. Let $t$ be the number of hours which the train has been on its journey from Euston, and $s$ be the number of miles passed over. Then $s$ is a function of $t$ , i.e. is the variable value corresponding to the variable argument $t$ .

If we know the circumstances of the train's run, we know $s$ as soon as any special value of $t$ is given. Now, miracles apart, we may confidently assume that $s$ is a continuous function of $t$ . It is impossible to allow for the contingency that we can trace the train continuously from Euston to Bletchley, and that then, without any intervening time, however short, it should appear at Rugby. The idea is too fantastic to enter into our calculation: it contemplates possibilities not to be found outside the Arabian Nights; and even in those tales sheer discontinuity of motion hardly enters into the imagination, they do not dare to tax our credulity with anything more than very unusual speed. But unusual speed is no contradiction to the great law of continuity of motion which appears to hold in nature. Thus light moves at the rate of about $190, 000$ miles per second and comes to us from the sun in seven or eight minutes; but, in spite of this speed, its distance travelled is always a continuous function of the time.

It is not quite so obvious to us that the velocity of a body is invariably a continuous function of the time. Consider the train at any time $t$ : it is moving with some definite velocity, say $v$ miles per hour, where $v$ is zero when the train is at rest in a station and is negative when the train is backing. Now we readily allow that $v$ cannot change its

value suddenly for a big, heavy train. The train certainly cannot be running at forty miles per hour from 11.45 a.m. up to noon, and then suddenly, without any lapse of time, commence running at $50$ miles per hour. We at once admit that the change of velocity will be a gradual process. But how about sudden blows of adequate magnitude? Suppose two trains collide; or, to take smaller objects, suppose a man kicks a football. It certainly appears to our sense as though the football began suddenly to move. Thus, in the case of velocity our senses do not revolt at the idea of its being a discontinuous function of the time, as they did at the idea of the train being instantaneously transported from Bletchley to Rugby. As a matter of fact, if the laws of motion, with their conception of mass, are true, there is no such thing as discontinuous velocity in nature. Anything that appears to our senses as discontinuous change of velocity must, according to them, be considered to be a case of gradual change which is too quick to be perceptible to us. It would be rash, however, to rush into the generalization that no discontinuous functions are presented to us in nature. A man who, trusting that the mean height of the land above sea-level between London and Paris was a continuous function of the distance from London, walked at night on Shakespeare's

Cliff by Dover in contemplation of the Milky Way, would be dead before he had had time to rearrange his ideas as to the necessity of caution in scientific conclusions.

It is very easy to find a discontinuous function, even if we confine ourselves to the 21 simplest of the algebraic formulæ. For example, take the function $y = \frac{1}{x}$ , which we have already considered in the form $p = \frac{1}{v}$ , where $v$ was confined to positive values. But

now let x have any value, positive or negative. The graph of the function is exhibited in [fig.]21. Suppose x to change continuously from a large negative value through a numerically decreasing set of negative values up to