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February 19th, 2012, 06:05 PM   #1
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is this property unique to linear polynomial functions?

Hi guys,
first post;D

I'm wondering if the following property is unique to linear functions:



I'm pretty sure it is, but I'm not sure how to go about proving it. I'm not asking for a proof, but simply to be pointed in the right direction.

Thanks!

noob
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February 19th, 2012, 06:13 PM   #2
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Can you state exactly what you mean (without using words such as "linear" and "polynomial")?
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February 19th, 2012, 06:22 PM   #3
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Re:

Quote:
Originally Posted by skipjack
Can you state exactly what you mean (without using words such as "linear" and "polynomial")?
Ok. By these terms, I mean a function that obeys

, where C is some constant.
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February 19th, 2012, 07:31 PM   #4
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Re: is this property unique to linear polynomial functions?

hey guys! Found it

http://retro.seals.ch/digbib/fr/view?ri ... &id2=&id3=
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February 20th, 2012, 09:59 AM   #5
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It's obvious that you can use a substitution to get f(x + y) ? f(x) + f(y). However, it wasn't clear that you would be happy with any function that obeys that equation, without necessarily being continuous (or having any other property that would imply its continuity). Adding on such extra information allows it to be proved that f(x) ? ax, where a is a constant.

Note that the terms "linear function" and "linear polynomial" are often used for functions of the form f(x) = ax + b, which do not satisfy the above equation unless b is zero.
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