Absolute of a function equal to zero

Question

How would I prove the following theorem:

if $\sup \{ |f (x)| :x>0\} = 0$

this implies

$|f (x)| = 0$ for all real number $x$.

Answer 1

What do you know about $f$? The statement is certainly not true for continuous functions on $\mathbb{R}$. A counter-example: $$ f(x) = |x| -x $$