contestada

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.

Respuesta :

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

[tex]0\le g(a)\le g(c)\le g(b)\le0[/tex]

which means we must have [tex]g(c)=0[/tex], or [tex]f(c)-c=0[/tex], or equivalently [tex]f(c)=c[/tex].

Otras preguntas