# Does the following relation always hold?

Given two functions $$f_1(x)=g_1(x)+h(x)$$ and $$f_2(x)=g_2(x)+h(x)$$ I know that $f_1(x)$ and $f_2(x)$ are monotone increasing. If $g_2(x)<g_3(x)<g_1(x)$, then is it true that $$f_3(x)=g_3(x)+h(x)$$ is also monotone increasing?

EDIT: I forgot to mention that $h(x)$ is also monotone increasing.

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With only those assumptions, a counterexample on $I=[0,1]$ is $h(x)=0$, $f_1(x)=2+x$, $f_2(x)=-2+x$, $f_3(x)=1-x$. – Sebastien B Feb 3 '13 at 11:36
@SebastienB I see. I forgot to put the condition on $h(x)$ otherwise it seems trivial just with omiting it via assigning $0$. I am editing. – Seyhmus Güngören Feb 3 '13 at 11:39
With $I=[0,1]$, $h(x)=x/2$, $g_1(x)=2+x$, $g_2(x)=-2+x$, $g_3(x)=1-x$. – Sebastien B Feb 3 '13 at 11:49
@SebastienB thank you very much for the counterexample on $I=[0,1]$. Is it also true when $I=\mathbb{R}$? – Seyhmus Güngören Feb 3 '13 at 12:01

For $I=\mathbb{R}$, a counter example is $g_1=2,g_2=-2,g_3(x)=\frac{3}{2}\sin x,h(x)=x$