1908-9. j Reversibility of Order of Partial Differentiation. 161 
violates the result of Cor. 4 owing to the fact that the first derivates with 
respect to x are not continuous with respect to y . 
In both these examples the appearance of infinite values for certain of 
the first derivates disturbs the existence of certain of the second derivates. 
The following example is, for this reason, perhaps more striking. 
Example 3. — 
f( x 5 v) = v - x > if * < v , 
= x 2 — ?/ 2 , if x ^ y . 
The region considered is the square (0,0; 1,1). 
Here f(x , y) is a continuous function of the ensemble (x , y); its 
derivates are, however, discontinuous. 
.f-x =f x = - 1 , if x L y , 
= 2x if x > y . 
Therefore the right-hand derivates with respect to y of the left-hand 
derivate with respect to x is zero everywhere, but the left-hand derivate 
with respect to y , though zero except on the line x = y , has here the value 
+ oo , since it is the limit of 
- 1 - 2x 
T 
Again, 
f +x =/ +x = - 1 , if x <y i 
— 2x if x \ y . 
Therefore the left-hand derivate with respect to y of the right-hand 
derivate with respect to x is zero everywhere, but the right-hand derivate 
with respect to y , though zero except on the line x = y , has here the value 
+ x . Hence 
L = 0, 
while, with an obvious notation for the upper bounds, 
= 9 j IT —y, —x H+i/, + X . 
§ 16. Of course, if all the derivates with respect to y of some derivate 
of f{x , y) with respect to x are finite at any point, that first derivate is 
bound to be continuous at the point with respect to y . Hence, as a special 
case of Cor. 4, we have the following : — 
If f(x , y) is a continuous function of x and of y , and if the derivates 
with respect to one variable of one or more derivates of f(x , y) with respect 
to the other variable are all finite in a neighbourhood of a point P , they 
cdl have the same plane associated limiting functions <fi and \h cd P , and 
lie between them. 
VOL. XXIX. 
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