Hello

function $\displaystyle f :[0,1]\rightarrow [0,1]$ is continuous. Also range of $\displaystyle f$ is $\displaystyle [0,1]$. I have to use intermediate value theorem to prove that $\displaystyle f$ must have a fixed point in it domain. Now let's define $\displaystyle g(x) = f(x) - x$. So $\displaystyle g(0) = f(0)$ and $\displaystyle g(1) = f(1)-1$. Now since $\displaystyle f$ is continuous on $\displaystyle [0,1]$, $\displaystyle g$ is also continuous on $\displaystyle [0,1]$. Consider the product $\displaystyle g(0)g(1)$. We have three possibilities here.

Case 1)$\displaystyle g(0)g(1) > 0$

So, we have two sub cases here. Sub case 1 is $\displaystyle g(0) > 0$ and $\displaystyle g(1) >0$. $\displaystyle g(0) > 0\Rightarrow f(0) >0$ and $\displaystyle g(1) > 0 \Rightarrow f(1) > 1$. But since range of $\displaystyle f$ is $\displaystyle [0,1]$, $\displaystyle f(1)\ngtr 1$. So this sub case is not possible. Consider the second sub case here. $\displaystyle g(0) <0$ and $\displaystyle g(1) <0$. $\displaystyle g(0) <0 \Rightarrow f(0) < 0$. Since range of $\displaystyle f$ is $\displaystyle [0,1]$, $\displaystyle f(0) \nless 0$, hence the second sub case is also not possible. Which means, case 1 is impossible for function $\displaystyle f$. Let's go to the case two

Case 2)$\displaystyle g(0)g(1) < 0$

First sub case is that $\displaystyle g(0) < 0$ and $\displaystyle g(1) > 0$. Since $\displaystyle g(0) < 0 < g(1)$, from intermediate value theorem, $\displaystyle \exists~ c \in (0,1)$ such that $\displaystyle g(c) = 0$, which means we have $\displaystyle f(c) = c$ and $\displaystyle c \in \text{Dom}(f)$. So $\displaystyle c$ is the fixed point of function $\displaystyle f$. Second sub case is $\displaystyle g(0) >0$ and $\displaystyle g(1) < 0$. The proof will be on similar lines. Now let's go to the last case

Case 3)$\displaystyle g(0)g(1) = 0$

Here either $\displaystyle g(0) = 0$ or $\displaystyle f(1)=1$. So either $\displaystyle f(0) = 0$ or $\displaystyle f(1) = 1$. Since $\displaystyle 0$ and $\displaystyle 1$ are in domain of $\displaystyle f$, we proved that there is a fixed point of $\displaystyle f$.

Since all cases are exhausted, we proved that $\displaystyle f$ must have a fixed point.

Is my proof correct ?

Thanks