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## Homework Statement

Suppose f(x)f(y)=f(x+y) for all real x and y,

(a) Assuming that f is differentiable and not zero, prove that

[tex]f(x) = e^{cx}[/tex]

where c is a constant.

(b) Prove the same thing, assuming only that f is continuous.

## Homework Equations

## The Attempt at a Solution

(a) The given information implies that f(x)^2 = f(2 x) and furthermore that f(x)^n = f(n x) for any natural number n. We can differentiate that to obtain f(x)^(n-1) f'(x) = f'(n x) for any natural number n.

It is also not hard to show that f'(x) f(y) = f'(y) f(x) for all real x and y.

We can take y = 0 to find that f(0) = 1 (I have used the fact that f is nonzero here).