Show that $f$ is continuous and hence of the form $f(x)=cx$

Asked Mar 12 '16 at 06:33

Active Mar 12 '16 at 07:36

Viewed 198 times

Let $f:\Bbb R\to \Bbb R$ be additive i.e. satisfies $f(x+y)=f(x)+f(y)$ for all $x$, $y$. and Lebesgue measurable.

Show that $f$ is continuous and hence of the form $f(x)=cx$.

In order to show that $f$ is continuous at a point $a$ we have to show that $|f(x)-f(a)|<\epsilon $ whenever $|x-a|<\delta$.

But I can't find any way to proceed here. Please give some hints.

edited Mar 12 '16 at 07:36

Martin Sleziak

asked Mar 12 '16 at 06:33

Learnmore

2

You may want to check http://math.stackexchange.com/questions/423492/overview-of-basic-facts-about-cauchy-functional-equation – Macavity Mar 12 '16 at 07:23
4

See http://math.stackexchange.com/questions/45861/show-that-fxy-fxfy-implies-f-continuous-leftrightarrow-f-measur, http://math.stackexchange.com/questions/359183/measurable-cauchy-function-is-continuous or http://math.stackexchange.com/questions/386841/prove-that-if-a-particular-function-is-measurable-then-its-image-is-a-rect-line – Martin Sleziak Mar 12 '16 at 07:39

0 Answers0