Does the following relation always hold?

Question

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.

Answer 1

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