Friday, January 11, 2019

calculus - How to show that If f(x+y)=f(x)f(y) then f(x)geq0





I would appreciate if somebody could help me with the following problem:



Q: Suppose f:RR is a continuous function and f(x+y)=f(x)f(y). Then f(x)0


Answer



If f vanishes at one point then it vanishes everywhere. So one solution is f(x)=0 for all x. Otherwise f must be non-zero everywhere. Now it is easy to see that if f(x)<0 then f(2x)=f(x)f(x)>0 and hence f changes sign and vanishes somewhere by IVT. Hence we must have f(x)>0 for all x.


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