Let $f:[a,b] \rightarrow \mathbb{R}$ a continuous function and consider points $y_1$ and $y_2$ from $[a,b]$ ($y_1 \neq y_2$).
(i) Supose that $f(y_1) \geq y_1$ and $f(y_2) \leq y_2$ and prove that there is such a $\gamma$ between $x_1$ and $y_2$ that $f(\gamma) = \gamma$.
I know that i can use the mean value for some $h(x) = f(x) - x$. I can show for $f(y_ 1) = y_ 1$ and $f(y_ 2) = y_ 2$ but for the other case ($f(y_1) > y_1$ and $f(y_2) < y_2$) i need some help.
(ii) Supose that $f(y_1) = y_2$ and $f(y_2) = y_1$ and prove that there is such a $\gamma$ between $x_1$ and $y_2$ that $f(\gamma) = \gamma$.
this case is the same as i suposed to do in the case ($f(y_ 1) = y_ 1$ and $f(y_ 2) = y_ 2$)?