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

150 INTRODUCTION TO MATHEMATICS
higher mathematics, but we need not concern
ourselves further about them here.
The most important division between functions is that between continuous and discontinuous functions. A function is continuous
when its value only alters gradually for
gradual alterations of the argument, and is
discontinuous when it can alter its value by
sudden jumps. Thus the two functions x +
and 1-x, whose graphs are depicted as
straight lines in fig. 14 of Chapter IX, are continuous functions, and so is the function 1,
depicted in Chapter II, if we only think of
positive values of v. But the function depicted in fig. 20 of this chapter is discontinuous since at the values x = 1, x =2, etc., of its
argument, its value gives sudden jumps.
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