Thursday, September 14, 2017

real analysis - f(a+b)=f(a)+f(b). Prove that f(x)=Cx, where C=f(1)

A question from Introduction to Analysis by Arthur Mattuck:




Suppose f(x) is continuous for all x and f(a+b)=f(a)+f(b) for all a and b. Prove that f(x)=Cx, where C=f(1), as follows:



(a)prove, in order, that it is true when x=n,1n and mn, where m,n are integers, n0;



(b)use the continuity of f to show it is true for all x.



I can show the statement is true when x=n. As for x=1n,mn, I don't know how.

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