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 $\displaystyle f$ is continuous on $\displaystyle [0,1]$ and $\displaystyle f(x)\epsilon[0,1]$ for every $\displaystyle x\epsilon[0,1]$.
If $\displaystyle f(0)=0$ or $\displaystyle f(1)=1$, the theorem is proved. So we try to prove the theorem assuming
$\displaystyle f(0)>0$ and $\displaystyle f(1)<1$.
Let $\displaystyle g(x)=f(x)-x$ for all $\displaystyle x\epsilon[0,1]$. Hence $\displaystyle g(0)>0$ and $\displaystyle g(1)<0$ and $\displaystyle g$ is continuous on $\displaystyle [0,1]$, that is, $\displaystyle 0$ is an intermediate value of $\displaystyle g$ on $\displaystyle [0,1]$. Hence by intermediate value theorem, there exists a point
$\displaystyle c\epsilon(0,1)$ such that $\displaystyle g(c)=0$ --which means $\displaystyle f(c)=c.$ Hence the prrof.
EDIT: $\displaystyle c$ is equivalent to $\displaystyle x_0$