160 Proceedings of the Royal Society of Edinburgh. [Sess. 
theorems of the paper “ On Derivates and the Theorem of the Mean,” used 
in proving those of the present paper, do not in general hold if we omit 
the condition of continuity. 
Thus, for instance, denoting by w(x) the function graphically represented 
by the bisector of the angle between the axes of x and y from the origin 
to x = i both inclusive, and by the line perpendicular to this through the 
point (1,0) from x= \ not inclusive to x = l inclusive (fig. 1), the function 
w(x) does not obey the theorems of the paper quoted, although it is 
continuous except only at « = In fact, the left-hand derivate is +1 
from x = 0 to x — J , and is — 1 afterwards. But the right-hand derivate, 
0 12 1 
u f$ 3 1 
Fig. 1. 
though it agrees with the left-hand one, except at the discontinuity x = ^ , 
has there the value + oo . 
From this function it is easy to construct a function of x and y which 
violates the result of the preceding theorem, owing to the fact that it is 
not a continuous function of x , or that its derivates with respect to x are 
discontinuous functions of y . 
Example 1 . — Let 
/(a, y) = yw{x). 
Here the left-hand derivate with respect to x is always either y or — y , 
so that the left-hand derivates with respect to y of the left-hand derivate 
with respect to x are always finite, being either 1 or — 1 . But the right- 
hand derivates of the same derivate, though agreeing with its left-hand 
derivates except on the ordinate x = -J,have there the value -foo. Thus 
the upper bound is in the case of one repeated derivate finite and in 
another infinite. 
Example 2. — In the preceding example f(x , y) was itself discontinuous 
with respect to x . If we write * 
f(x , y) = xw(y) , 
we get a function which is continuous with respect to x , and which yet 
