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Question Number 132932 by bobhans last updated on 17/Feb/21

$$Findf\left(x\right)suchthatf\left(2x\right)=f\left(x\right)$$

Answered by Olaf last updated on 17/Feb/21

$$f\left(2x\right)=f\left(x\right)$$$$\Rightarrow f\left(x\right)=f\left(\frac{x}{2}\right)=f\left(\frac{x}{4}\right)=...f\left(\frac{x}{2^n}\right)...$$$$f\left(x\right)=\underset{n\to \mathrm{\infty }}{\lim }f\left(\frac{x}{2^n}\right)=f\left(0\right)=\mathrm{constant}$$$$\mathrm{If}f\left(0\right)\mathrm{exists},f\mathrm{is}\mathrm{a}\mathrm{constant}\mathrm{function}$$$$\mathrm{and}f\left(x\right)=f\left(0\right)$$

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