Hardest problem i have enountered so far.FUNTIONS

[FUGU]

f,g : [0,1]-->[0,1] both contineous funtions
f(g(x))=g(f(x))

To solve:
to prove that there exists a numer "c" where cis contained within (0,1)
where f(c)=g(c).


NOTE. I have used the Darboux property but i dont know how to continue next ,as the functions are not identical.

 

[daon2]

f,g : [0,1]-->[0,1] both contineous funtions
f(g(x))=g(f(x))

To solve:
to prove that there exists a numer "c" where cis contained within (0,1)
where f(c)=g(c).


NOTE. I have used the Darboux property but i dont know how to continue next ,as the functions are not identical.

It isn't true. Take f(x)=x, g(x)=x^2.

 

[FUGU]

It isn't true. Take f(x)=x, g(x)=x^2.

not for every number.It says to prove that there exists at least one

 

[daon2]

not for every number.It says to prove that there exists at least one

If for every number c in (0,1), f(c) and g(c) are different, how do you suppose you can find at least one c where they are the same?

 

[FUGU]

If for every number c in (0,1), f(c) and g(c) are different, how do you suppose you can find at least one c where they are the same?

it says that "c " is a number in (0,1) .Not every single number. "c" is just ONE number

 

[Ishuda]

It isn't true. Take f(x)=x, g(x)=x^2.

not for every number.It says to prove that there exists at least one

It isn't true for any number in (0,1). If it were, then
f(c) = c = g(c) = c²
which only has solutions 0 and 1 neither of which are in (0,1).