Is $f : [0,\infty]\rightarrow [0,\infty]$ which is continuous and bounded has a fixed point

Question

Question is to check if :

$f : [0,\infty]\rightarrow [0,\infty]$ which is continuous and bounded has a fixed point.

I have first of all considered boundedness.

So, $f(x)$ should not have $x$ as multiple i mean $f(x)$ should not have a polynomial factor.

I took $f(x)=\frac{1}{g(x)}$ and $g(x)$ should not have positive root (assuming it to be polynomial)

for fixed point, $\frac{1}{g(x)}=x$, we have $x.g(x)=1$

for simple $g(x)$ i have ended up with concluding that $f(x)$ have fixed point.

I do not have any clear idea how to porve this.. So, I have written what are all i have thought could be helpful...

please give some hint to crack this.

Thank you..

Answer 1

For fixed point $x_1$, we have $f(x_1) = x_1$ (that is what fixed point means). Let's say $0$ is not a fixed point. Then $f(0) \gt 0$ (it can't be less, by definition of $f$, and we assume it's not equal). so there are values of $x$ for which $f(x) > x$; that is, the function $g(x) = f(x) - x$ is positive.

Now, $f(x)$ is bounded, so there is some $K$ such that $f(x) < K$ for all $x$. Especially, $f(K) < K$, so $g(K) < 0$.

Now, the function $g(x)$ is continuous, and it is positive at some points (for instance, at $0$), and it is negative at some points (for instance, at $K$), so it has to be $0$ at some points (specifically between $0$ and $K$). That's your fixed point.