Originally Posted by

**derek walcott** PROBLEM:

Let a and b be real numbers with a<b. Suppose that f:[a,b]-->R is a continuous function that satisfies f([a,b]) is a set of [a,b].

Prove that f has a fixed point.

{The hint is: g=f(x)-x and use IVT}

ATTEMPT AT PROOF:

so using IVT if there is a and b then there must be a c so, a<c<b

which means f(a)<f(c)<f(b) or f(a)>f(c)>f(b). if f([a,b]) is a set of [a,b] then at most they are the same interval. if f is a fixed point then f(x)-x = 0 so, g=0.

OKay here is where i am stuck ..... if g(a)>0 and g(b)<0 then there is a fixed point and if g(a)<0 and g(b)>0 there is a fixed point but is that all they can ever be? .... is there an instant where g(a)<0 and g(b)<0 and still be existent with a fixed point in this interval.

But g(a)< 0 and g(b)< 0 is not possible.