Question

Suppose f is a real valued function mapping the reals to the reals for which f(x+y) = f(s)f(y). Prove that f having a limit at zero implies f has a limit everywhere and that this limit is one if f is not identically zero

Answer #1

such that

We are given that exists

Then

Thus, so that

If then we have

Which means so that the function is identically zero which not possible

So we must have and so

So that we must have if the function is not identically zero

Moreover, the limit exists everywhere because

Which is as

We note that a finite quantity and from above

Thus, is a finite quantity and so the limit exists for all

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