lean_workbook_plus_27916

Because $f$ is strictly increasing and $f\left({\left[{0,\,1}\right]}\right)=\left[{0,\,1}\right]$ , then $f(0)=0$ and $f(1)=1$ . If $g(0)=0$ or $g(1)=1$ , then $g(0)=0=f(0)$ or $g(1)=1=f(1)$ . If $g(0)\neq0$ and $g(1)\neq1$ , because $g\left({\left[{0,\,1}\right]}\right)=\left[{0,\,1}\right]$ , then $g(0)>0$ and $g(1)<1$ . For the continuous in $\left[{0,\,1}\right]$ function $g$ there is $^{(*)}$ $x_0\in\left({0,\,1}\right)$ , such that $g(x_0)=x_0$ .