Absolute value of functions

Question

Let $f,g,h:\mathbb{R}\to\mathbb{R}$, then is it true the following: $$ fh=gh\Longrightarrow f|h|=g|h|? $$ where $|z|=z^++z^-$ and $z^+,z^-$ are respectively the positive and negative part of a function $z$. I don't think itis true, I have tried to find some counterexamples using indicator functions, but I have not managed to disprove it (or to prove it).

Answer 1

It is true. Just consider the cases $h>0, h<0$ and $h=0$ to prove it.

Answer 2

It is not true if $fh$ stands for composition $f\circ h$.

Let $h$ be prescribed by $x\mapsto1$ if $x<0$ and $x\mapsto -2$ otherwise.

Let $f$ be the identity.

Let $g$ be prescribed by $g(1)=1$, $g(-2)=-2$ and $x\mapsto 0$ if $x\notin\{1,-2\}$.

Then $f\circ h=g\circ h$ but not $f\circ|h|=g\circ|h|$.

Note that e.g. $f\circ|h|(0)=f(|h|(0))=f(2)=2$ and $g\circ|h|(0)=g(|h|(0))=g(2)=0$.