**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. **

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