DERIVATIVES AND INTEGRALS 
188 
[VI 
is sufficient to consider the curve formed by two straight lines 
meeting to form an angle (Fig. 43). 
The reader will verify at once that in 
this case {<f>(x + h) — (f)(x)}/h has the 
limit tana as h-*-0, if x<x 0 , but the 
limit tan /3 if x > x 0 . And 
{<£ (x 0 + h) - <j> (x 0 )]/h 
has no limit as A^O. In fact it has Fig. 43. 
the limit tan /3 if h-*~0 by positive 
values and the limit tana if A-*-0 by negative values. 
This is of course one of the cases in which a curve might reasonably be 
said to have two directions at a point. But the following example, although 
more difficult, shows conclusively that there are cases in which a continuous 
curve cannot be said to have either one definite direction or several directions 
at one of its points. Draw the graph of the function x sin (1/#) (Fig. 16, 
Ch. II). The function is not defined for # = 0 and so is discontinuous for 
x=0. But (Exs. xxxviii. 17) the function defined by the equations 
(f> (x) = x sin (1 jx) (a?4= 0), (f) (x)=0 {x = 0) 
is continuous for .r = 0: and the graph of this function, if it could be drawn 
completely (and we can draw it quite adequately enough to obtain a general 
idea of its appearance) would satisfy our common sense intuition of what a 
continuous curve should be like*. 
But 4>{x) has no derivative for x=0. For </>'(0) would be, by definition, 
lim {(f> (h) - $ (0)}//i or lim sin (1 Jh) and this limit does not exist. The reader, 
on studying the figure carefully, will probably agree that the curve does not 
look as if it had a tangent or a definite direction at the origin. 
It has even been shown that a function of x may be continuous and yet 
have no derivative for any value of x. But the proof of this is much more 
difficult; the reader who is interested in the question may be referred to 
Bromwich’s Infinite Series, pp. 490-1, or Hobson’s Theory of Functions 
of a Real Variable, pp. 620-5. 
(3) Rates of Variation. There is another general point of 
view from which the notion of the derivative of a function </>(#) 
may be considered. Let us suppose that OM, measured along 
OX, represents the value of x, and ON, measured along OY, 
represents the value of y = f>{x): so that the corresponding point 
on the graph of </>(&•) is obtained by completing the rectangle 
OMPN. Further, let us imagine the variation of x as taking 
* No doubt it is a somewhat peculiar curve, but there is obviously no breach of 
continuity. The apparently arbitrary assignment of the value 0 to (p (x) for a; = 0 
is in reality natural enough—it merely amounts to the filling in, so to say, of a 
single point previously missing in the curve y = a;sin (1 jx).