Give-a-function-f-R-0-continous-on-R-and-such-that-f-x-y-f-x-f-y-a-Prove-f-0-1-b-Let-h-x-ln-f-x-Prove-that-h-x-y-h-x-h-y-c-Find-all-the-function-f-such-that-problem-re

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Question Number 199167 by tri26112004 last updated on 28/Oct/23

$$Giveafunction$$$$f:R\to (0;+\mathrm{\infty })continousonRandsuchthat$$$$f(x+y)=f\left(x\right).f\left(y\right)$$$$a.Provef\left(0\right)=1$$$$b.Leth\left(x\right)=ln\left[f\left(x\right)\right].Provethat:$$$$h(x+y)=h\left(x\right)+h\left(y\right)$$$$c.Findallthefunctionfsuchthatproblemrequest$$$$$$$$$$

Answered by AST last updated on 28/Oct/23

$$a.f\left(0\right)={\left[f\left(0\right)\right]}^2\Rightarrow f\left(0\right)=0or1butf\left(0\right)\ne 0\Rightarrow f\left(0\right)=1$$$$b.h(x+y)=In\left[f(x+y)\right]=In[f\left(x\right).f\left(y\right)]$$$$=In\left[f\left(x\right)\right]+In\left[f\left(y\right)\right]=h\left(x\right)+h\left(y\right)$$$$c.e^{h\left(x\right)}=f\left(x\right)forcontinuoush\left(x\right)$$$$Orgenerally,a^{h\left(x\right)}forcontinuoush\left(x\right)$$$$wherea>1$$

Commented by tri26112004 last updated on 28/Oct/23

$$Thankyousomuch!$$$$$$

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