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Question Number 4873 by prakash jain last updated on 18/Mar/16
$$\mathrm{If}\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{Continous}\:\mathrm{and}\:\mathrm{odd}\:\mathrm{function}. \\ $$$$\mathrm{Then}\:\mathrm{is}\:{f}\left(\mathrm{0}\right)=\mathrm{0}? \\ $$
Commented by prakash jain last updated on 18/Mar/16
$$\mathrm{Thanks}.\:\mathrm{That}\:\mathrm{is}\:\mathrm{what}\:\mathrm{I}\:\mathrm{thought}\:\mathrm{but}\:\mathrm{couldn}'\mathrm{t}\:\mathrm{think}\:\mathrm{of} \\ $$$$\mathrm{proof}. \\ $$
Commented by Rasheed Soomro last updated on 19/Mar/16
$${f}\left(\mathrm{0}\right)=−{f}\left(\mathrm{0}\right)\Rightarrow{f}\left(\mathrm{0}\right)=\mathrm{0}\:? \\ $$$${How}\:{f}\left(\mathrm{0}\right)=\mathrm{0}\:? \\ $$
Commented by 123456 last updated on 19/Mar/16
$${f}\left(\mathrm{0}\right)=−{f}\left(−\mathrm{0}\right)=−{f}\left(\mathrm{0}\right) \\ $$$${f}\left(\mathrm{0}\right)+{f}\left(\mathrm{0}\right)=−{f}\left(\mathrm{0}\right)+{f}\left(\mathrm{0}\right) \\ $$$$\mathrm{2}{f}\left(\mathrm{0}\right)=\mathrm{0} \\ $$$${f}\left(\mathrm{0}\right)=\mathrm{0}/\mathrm{2}=\mathrm{0} \\ $$
Commented by Rasheed Soomro last updated on 19/Mar/16
$$\mathbb{T}\boldsymbol{\mathrm{h}\alpha\mathrm{n}}\Bbbk\boldsymbol{\mathrm{s}}! \\ $$
Commented by 123456 last updated on 18/Mar/16
$$\mathrm{since}\:{f}\:\mathrm{is}\:\mathrm{continuous} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{f}\left({x}\right)={f}\left(\mathrm{0}\right) \\ $$$${f}\left({x}\right)=−{f}\left(−{x}\right) \\ $$$${x}=\mathrm{0} \\ $$$${f}\left(\mathrm{0}\right)=−{f}\left(\mathrm{0}\right)\Rightarrow{f}\left(\mathrm{0}\right)=\mathrm{0} \\ $$