Linear ProofDate: 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. k->infinity 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 http://mathforum.org/dr.math/ |
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