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 February 19th, 2012, 06:05 PM #1 Newbie   Joined: Feb 2012 Posts: 3 Thanks: 0 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: $f(x_1 - x_2)= f(x_1) - f(x_2)$ 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
 February 19th, 2012, 06:13 PM #2 Global Moderator   Joined: Dec 2006 Posts: 20,634 Thanks: 2080 Can you state exactly what you mean (without using words such as "linear" and "polynomial")?
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

$f(x)= C \cdot x^{n}, n=1$, where C is some constant.

 February 19th, 2012, 07:31 PM #4 Newbie   Joined: Feb 2012 Posts: 3 Thanks: 0 Re: is this property unique to linear polynomial functions? hey guys! Found it http://retro.seals.ch/digbib/fr/view?ri ... &id2=&id3=
 February 20th, 2012, 09:59 AM #5 Global Moderator   Joined: Dec 2006 Posts: 20,634 Thanks: 2080 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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