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letsw-f-x-y-f-x-f-y-f-1-1-f-x-f-1-x-1-x-0-proof-that-f-x-x-x-R-

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Question Number 1251 by 123456 last updated on 18/Jul/15
letsw f(x+y)=f(x)+f(y) f(1)=1 f(x)f((1/x))=1,x≠0 proof that f(x)=x∀x∈R
letswf(x+y)=f(x)+f(y)f(1)=1f(x)f(1x)=1,x≠0proofthatf(x)=x∀x∈R
Commented by prakash jain last updated on 18/Jul/15
Cauchy′s Equation f(x+y)=f(x)+f(y)⇒f(x)=ax Given f(x)f((1/x))=1⇒ax∙(a/x)=1⇒a=±1 f(x)=x or f(x)=−x A condtion f(1)=1 is required to prove f(x)=x
Cauchy′sEquationf(x+y)=f(x)+f(y)⇒f(x)=axGivenf(x)f(1x)=1⇒ax⋅ax=1⇒a=±1f(x)=xorf(x)=−xAcondtionf(1)=1isrequiredtoprovef(x)=x

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