Intuitively this is obvious by graphing g(x) = x on [0,1] and seeing since f is continuous it has to intersect with g at some point. But I spent a long time and cannot figure out how to prove this.
Intuitively this is obvious by graphing g(x) = x on [0,1] and seeing since f is continuous it has to intersect with g at some point. But I spent a long time and cannot figure out how to prove this.
It is the well-known fixed point theorem. Suppose is continuous on and for every .
If or , the theorem is proved. So we try to prove the theorem assuming
and .
Let for all . Hence and and is continuous on , that is, is an intermediate value of on . Hence by intermediate value theorem, there exists a point
such that --which means Hence the prrof.
EDIT: is equivalent to