Question: Determine all the functions f, which are everywhere differentiable and satisfy
for all real x and y with . —- Equation I
Let satisfy Equation I above. Differentiating Equation I partially with respect to each of x and y, we obtain
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.