Question

Suppose f : [a, b] → [a, b] is a continuous function. Prove that it has a fixed point x (that is, a point x such that f(x) = x).

Answer 1

here we consider a function g(x)=f(x)-x.then we prove g(a)0 and g(b)0.if g(a)=0 and g(b)=0 then f(a)=a and f(b)=b hence in this case a or b itself a fixed point x of f.so if g(a)>0 and g(b)<0 by intermediate value theorem there exist a x in (a,b) such that g(x)=0 so f(x)-x=0 hence f(x)=x.so there exist a fixed point x .