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April 11th, 2012, 11:15 AM   #1
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A homogeneous variable equation, not so easy!!!

Prove that if F(x+y)=F(x)+F(y) then F(z)=a*z .
(Do not involve any types of iteration please, )
Thank you
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April 11th, 2012, 01:13 PM   #2
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Re: A homogeneous variable equation, not so easy!!!

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Originally Posted by mathbalarka
Prove that if F(x+y)=F(x)+F(y) then F(z)=a*z .
(Do not involve any types of iteration please, )
Thank you
It is not easy, particularly since you need to assume continuity for it to hold.
The general idea is to first prove it for y as an integer multiple of x, using mathematical induction. This can then be extended to fractions. To have it hold for reals, you need continuity.
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