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
We are given that exists
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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