Question

Let [a,b] ⊆ ℝ be a close bounded interval, and let f : [a,b] → [a,b] be a function. suppose that f is continuous. prove that there is some c ∈ [a,b] such that f(c) =

c. the number c is called a fixed point of f.

Answer 1

Suppose
`f(a)\ge a`, so that
`f(a)-a\ge0`, and suppose
`f(b)\le b`, so that
`f(b)-b\le0`. Now consider the function
`g(x)=f(x)-x`.
`g` is clearly continuous. By the intermediate value theorem, we know there is some
`c\in[a,b]` such that


`0\le g(a)\le g(c)\le g(b)\le0`

which means we must have
`g(c)=0`, or
`f(c)-c=0`, or equivalently
`f(c)=c`.