Solving Functional Equation Homework

[zenos]

Homework Statement

Is the solution correct

Homework Equations

The Attempt at a Solution

all are in the file

 

[snipez90]

That does not look like a complete solution. For one, f(x) = 1 for all x and f identically equal to 0 are trivial solutions, and these can be cited by inspection. I'll keep trying things, but it would help if there were any additional assumptions on f, such as continuity perhaps?

 

[snipez90]

All right, here is a rough sketch for the case where f is continuous. As before, f identically equal to zero is a trivial solution. Now suppose there exists a real number c for which f(c) =/= 0. Then

$$f(x)f(c) = f(\sqrt{x^2 + c^2}) = f(-x)f(c).$$

This implies that f(x) = f(|x|) for all real x. Define $g(x) = f(\sqrt{x})$ for $x \geq 0$. Note that g satisfies Cauchy's exponential equation: g(x + y) = g(x)g(y) for $x,y \geq 0.$

Now see if you can complete the argument based off of the proof for Cauchy's exponential equation. For reference, attached is something I wrote awhile ago when I was still interested in functional equations.