Question: Determine all the functions f, which are everywhere differentiable and satisfy 

for all real x and y with . —- Equation I

Solution:

Let satisfy Equation I above. Differentiating Equation I partially with respect to each of x and y, we obtain

 Equation II

 Equation III

Eliminating common  terms in Equation II and Equation III, we deduce that

As the left side of Equation III depends only on x and the  right side only on y, each side of Equation III must be equal to a constant c. Thus, we have

and so, ,

for some constant d. However, taking in Equation III, we obtain , so that and

. Clearly, satisfies Equation I and so all solutions of Equation I are given by

where c is a constant.

Note: In Equation II, the is w.r.t. x and in Equation III, the is w.r.t. y, yet in equation II 

is the same as the corresponding in Equation III because the argument is symmetric w.r.t. x and y.