Cauchy's Problem

Question

I am looking at the Cauchy's functional equation here: http://en.wikipedia.org/wiki/Cauchy's_functional_equation.

Could someone help me on how to generalize the Cauchy's equation to $x \in \mathbb{R}$?

I know all the steps leading to the proof that the only possible functional equation $f(x)$ such that $f(x+y)=f(x)+f(y)$ for all $x \in \mathbb{Q}$ is the function $Cx$, where $x=f(1)$.

Thanks,

score 2 · Accepted Answer · answered Apr 24 '13 at 12:14

2

Hint: $f$ is continuous, for every irrational $s \in \mathbb Q'$ we have a sequence of rational numbers $r_n$ which converges to $s$, by continuity $f(r_n) \to f(s)$

answered Apr 24 '13 at 12:14

Mathematician

1,942

Thanks! I don't really understand how could a sequence of rational numbers that converge to an irrational number. Is it possible to provide an example? Wait... If this happens then $f(s)$ would be a also covered under the function $Cx$ where $C=f(1)$? – pseudo2013 Apr 24 '13 at 12:47
@pseudo2013 Consider $$\pi=3.14159\dots$$ Then the (infinite) sequence of rational numbers $$3, 3.1, 3.14, 3.141, 3.1415, 3.14159, \dots$$ converges to $\pi$. – Zubin Mukerjee Jan 04 '15 at 11:47
@pseudo2013 Without the assumption that $f$ is continuous, Cauchy's functional equation does not imply that $f$ is linear. There are some really strange functions that satisfy the functional equation and are definitely not linear. – Zubin Mukerjee Jan 04 '15 at 11:49

Cauchy's Problem

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