As many know, the intermediate value theorem states that given a function $f$ that is continuous on a closed interval $[a, b]$ with M a number between $f(a)$ and $f(b)$. Then there exists a number $c$ such that,
However I encountered contained a non-closed interval, an open one, it says: given $f$ and $g$ continuous functions on $[a,b]$ ($a<b$ of course) such that $g([a,b])=[a,b]$ (closed interval)) and $f([a,b])\subseteq]a,b[$, then show that there is at least one $c$ in $]a,b[$ (open interval) with $f(c)=g(c)$.
My problem is that I have no idea on how to apply the IVT in this case, can anyone give a hint please?
Edit All we need to show now, thanks to @TZakrevskiy, is that $$]x_1,x_2[\subseteq]a,b[$$ which is that which where i struggle Any hints?