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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: 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  
Newbie Joined: Feb 2012 Posts: 3 Thanks: 0  Re: Quote:
, 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? 
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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functions, linear, polynomial, property, unique 
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