Suppose f is continuous R to R and that f(x+y)=f(x)+f(y) for all x and y. Prove that f(x)=xf(1).

Hint: Prove the result for all Z and then for all Q.

Any help appreciated.

Jack

Printable View

- Nov 11th 2006, 01:14 PMJackStolermanAnother problem (additive function)
Suppose f is continuous R to R and that f(x+y)=f(x)+f(y) for all x and y. Prove that f(x)=xf(1).

Hint: Prove the result for all Z and then for all Q.

Any help appreciated.

Jack - Nov 11th 2006, 01:44 PMThePerfectHacker
The first question I ever asked on this forum!

(Long time ago, Look hier).

If a function is continous it must be a linear function thus proving your statement.

Otherwise, the axiom of choice can construct a non-linear solution :eek:

(This functional equation is one of my favorite math things I ever learned)