Hello

function is continuous. Also range of is . I have to use intermediate value theorem to prove that must have a fixed point in it domain. Now let's define . So and . Now since is continuous on , is also continuous on . Consider the product . We have three possibilities here.

Case 1)

So, we have two sub cases here. Sub case 1 is and . and . But since range of is , . So this sub case is not possible. Consider the second sub case here. and . . Since range of is , , hence the second sub case is also not possible. Which means, case 1 is impossible for function . Let's go to the case two

Case 2)

First sub case is that and . Since , from intermediate value theorem, such that , which means we have and . So is the fixed point of function . Second sub case is and . The proof will be on similar lines. Now let's go to the last case

Case 3)

Here either or . So either or . Since and are in domain of , we proved that there is a fixed point of .

Since all cases are exhausted, we proved that must have a fixed point.

Is my proof correct ?

Thanks