Suppose that $f$ is a function defined in $[a;b]$ to $[a;b]$ and continuous on $[a;b]$. The problem is I haven't the definition of the function, this is more abstract, but even if how can I prove that $f$ would have a fixed point?
In the plane: Every continuous function $f$ from a closed disk to itself has at least one fixed point.
This can be generalized to an arbitrary finite dimension:
In Euclidean space: Every continuous function from a closed ball of a Euclidean space to itself has a fixed point.
A slightly more general version is as follows:
Convex compact set: Every continuous function f from a convex compact subset K of a Euclidean space to K itself has a fixed point.
An even more general form is better known under a different name:
Schauder fixed point theorem: Every continuous function from a convex compact subset K of a Banach space to K itself has a fixed point.