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Linear Proof

Date: 11/07/2001 at 13:12:42
From: Emily Talbot
Subject: Challenging linear proof

Hi Dr. Math - 

We are asked to prove the following:

We say that f is linear provided that for every x, y in its domain, 
f(x+y) = f(x) + f(y). Show that if f is linear and continuous on R 
(the set of real numbers), then f is defined by f(x) = cx for some c 
belong to R. (It can be shown that not every linear function from R 
into R is continuous).

I'm not really sure how to begin this problem. Any help that you can 
offer me would be greatly appreciated. I understand the concept but I 
have a difficult time writing proofs. Thanks for your help.

Emily Talbot

Date: 11/07/2001 at 15:50:14
From: Doctor Rob
Subject: Re: Challenging linear proof

Thanks for writing to Ask Dr. Math, Emily.

Prove by induction that f(m*x) = m*f(x) for all positive integers m,
by using the linearity property.  Then use the linearity property
again to show that this implies that f(m*x) = m*f(x) for all
integers m.  Then for any nonzero integer n,

   f(x/n) = n*f(x/n)/n = f(n*[x/n])/n = f(x)/n.

Now set c = f(1), and use the above facts to show that for all
rational numbers m/n,

   f(m/n) = (m/n)*f(1) = c*(m/n).

So your statement is true if x is rational.

Now let x be irrational.  Let {y(k): k = 1, 2, ...} be a sequence of
rational numbers that converge to x,

      lim     y(k) = x.

For each k, f(y(k)) = c*y(k), because each y(k) is rational.  Now use
the continuity of f to conclude that f(x) = c*x for this case, too.

- Doctor Rob, The Math Forum   
Associated Topics:
College Number Theory
High School Number Theory

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