Question Number 4102 by prakash jain last updated on 28/Dec/15
$$\mathrm{Find}\:\mathrm{all}\:\mathrm{function}\:{f}:\mathbb{R}\rightarrow\mathbb{R} \\ $$$$\mathrm{such}\:\mathrm{that} \\ $$$${f}\left({x}+\mathrm{1}\right)={xf}\left({x}\right) \\ $$$${f}\:\left(\mathrm{0}\right)=\mathrm{1}\: \\ $$$${f}\left({x}\right)\neq\Gamma\left({x}\right) \\ $$
Commented by prakash jain last updated on 28/Dec/15
$$\Gamma\left({x}\right)\:\mathrm{is}\:\mathrm{gamma}\:\mathrm{function}. \\ $$
Commented by 123456 last updated on 28/Dec/15
$$\mathrm{i}\:\mathrm{think}\:\mathrm{that}\:\mathrm{if}\:\mathrm{function}\:\mathrm{need}\:\mathrm{to}\:\mathrm{be}\:\mathrm{continuous} \\ $$$$\mathrm{then}\:\Gamma\:\mathrm{is}\:\mathrm{the}\:\mathrm{only}\:\mathrm{function} \\ $$
Commented by Filup last updated on 29/Dec/15
$${x}=\mathrm{0} \\ $$$${f}\left(\mathrm{0}+\mathrm{1}\right)=\mathrm{0}{f}\left(\mathrm{0}\right) \\ $$$$\therefore{f}\left(\mathrm{1}\right)=\mathrm{0} \\ $$$$\Gamma\left(\mathrm{1}\right)=\mathrm{1} \\ $$$$\therefore{f}\left({x}\right)\neq\Gamma\left({x}\right)??? \\ $$
Commented by 123456 last updated on 29/Dec/15
$${f}\left(\mathrm{1}\right)=\mathrm{1}\:\mathrm{for}\:\mathrm{gamma}\:\mathrm{function} \\ $$
Commented by Filup last updated on 29/Dec/15
$$\mathrm{but}\:\mathrm{if}\:{f}\left({x}\right)=\Gamma\left({x}\right) \\ $$$${x}=\mathrm{1} \\ $$$${f}\left(\mathrm{1}\right)=\mathrm{0} \\ $$$$\Gamma\left(\mathrm{1}\right)=\mathrm{1} \\ $$$$\therefore{f}\left({x}\right)\neq\Gamma\left({x}\right) \\ $$$$\mathrm{I}\:{am}\:{confused} \\ $$
Commented by prakash jain last updated on 29/Dec/15
$$\mathrm{Find}\:{f}\:\mathrm{meeting}\:\mathrm{th}\:\mathrm{conditions}\:\mathrm{given}\:\mathrm{above}. \\ $$$$\mathrm{Functions}\:\mathrm{other}\:\mathrm{than}\:\Gamma\:\mathrm{function}. \\ $$