An introduction to mathematics, by A. N. Whitehead.

FUNCTIONS 151
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 cores 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