If-f-R-R-is-a-function-such-that-f-0-1-and-f-x-f-y-f-x-y-for-all-x-y-R-then-A-1-is-a-period-of-f-B-f-n-1-for-all-integers-n-C-f-n-n-for-all-integers-n-D-f-1-0-

Question Number 119801 by Ar Brandon last updated on 27/Oct/20

$$\mathrm{If}\:{f}:\mathbb{R}\rightarrow\mathbb{R}\:\mathrm{is}\:\mathrm{a}\:\mathrm{function}\:\mathrm{such}\:\mathrm{that}\:{f}\left(\mathrm{0}\right)=\mathrm{1}\:\mathrm{and}\:{f}\left(\mathrm{x}+{f}\left(\mathrm{y}\right)\right)= \\ $$$${f}\left(\mathrm{x}\right)+\mathrm{y}\:\mathrm{for}\:\mathrm{all}\:\mathrm{x},\:\mathrm{y}\in\mathbb{R},\:\mathrm{then} \\ $$$$\left(\mathrm{A}\right)\:\mathrm{1}\:\mathrm{is}\:\mathrm{a}\:\mathrm{period}\:\mathrm{of}\:{f} \\ $$$$\left(\mathrm{B}\right)\:{f}\left({n}\right)=\mathrm{1}\:\mathrm{for}\:\mathrm{all}\:\mathrm{integers}\:{n} \\ $$$$\left(\mathrm{C}\right)\:{f}\left({n}\right)={n}\:\mathrm{for}\:\mathrm{all}\:\mathrm{integers}\:{n} \\ $$$$\left(\mathrm{D}\right)\:{f}\left(−\mathrm{1}\right)=\mathrm{0} \\ $$

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