- #1
baconeater
- 1
- 0
Hello, think i have proved it but is the proof complete, is there any more i should do?
Let f:[0,1] --> [0,1], f be continuous, f(0)=0, f(1)=1
and let f(f(x)) = x, for all x in [0,1]
prove that f(x) = x.
(*) f(f(x)) = x, for all x in [0,1]
(**) f(0) = 0, f(1) = 1
Assume that f(a) = b>a, then it follows by (*) that f(f(a)) = f(b) = a<b.
Since f is continuous it follows by "Intermediate"- theorem that there exist
a point c where a< c <b such that f(c) = c.
Now we have that f(b) = a < c = f(c), since then there must exist a point d
where b< d <1 such that f(d) = c.
(if b=1 then f(b) = a <1, witch contradicts (**))
But then f(f(d)) = c < b witch is an contradiction!
Assume that f(a) = b < a, then it follows by (*) that f(f(a)) = f(b) = a>b.
Since f is continuous it follows by "Intermediate"- theorem that there exist
a point c where b< c <a such that f(c) = c.
Now we have that f(c) = c < a = f(b), since then there must exist a point d
where 0< d <b such that f(d) = c.
(if b=0 then f(b) = a > b, witch contradicts (**))
But then f(f(d)) = c < a witch is an contradiction!
Homework Statement
Let f:[0,1] --> [0,1], f be continuous, f(0)=0, f(1)=1
and let f(f(x)) = x, for all x in [0,1]
prove that f(x) = x.
Homework Equations
The Attempt at a Solution
(*) f(f(x)) = x, for all x in [0,1]
(**) f(0) = 0, f(1) = 1
Assume that f(a) = b>a, then it follows by (*) that f(f(a)) = f(b) = a<b.
Since f is continuous it follows by "Intermediate"- theorem that there exist
a point c where a< c <b such that f(c) = c.
Now we have that f(b) = a < c = f(c), since then there must exist a point d
where b< d <1 such that f(d) = c.
(if b=1 then f(b) = a <1, witch contradicts (**))
But then f(f(d)) = c < b witch is an contradiction!
Assume that f(a) = b < a, then it follows by (*) that f(f(a)) = f(b) = a>b.
Since f is continuous it follows by "Intermediate"- theorem that there exist
a point c where b< c <a such that f(c) = c.
Now we have that f(c) = c < a = f(b), since then there must exist a point d
where 0< d <b such that f(d) = c.
(if b=0 then f(b) = a > b, witch contradicts (**))
But then f(f(d)) = c < a witch is an contradiction!