陶哲轩实分析（上）9.7及习题-Analysis I 9.7

介值定理。很短但很有用。

Exercise 9.7.1

Since $f$ is a continuous function on $[a,b]$, by the maximum principle, there’s $x_1,x_2∈[a,b]$ such that
$M=f(x_1 ),\quad m=f(x_2 )$
If $x_1=x_2$, then let $c=x_1$ and the proof is over, assume $x_1<x_2$, then by Exercise 9.4.6, we have $f_{[x_1,x_2]}$ a continuous function on $[x_1,x_2 ]$, by Theorem 9.7.1, $∃c∈[x_1,x_2 ]⊆[a,b]$, s.t. $f(c)=y$.

Exercise 9.7.2

We let $F(x)=f(x)-x$, by Proposition 9.4.9 and Exercise 9.4.6, $F(x)$ is continuous on $[0,1]$, since $f$ has range $[0,1]$, we know that $f(0)≥0$ and $f(1)≤1$, thus
$F(0)=f(0)≥0,\quad F(1)=f(1)-1≤0$
Since $F(0)≥0≥F(1)$, by the Intermediate value theorem, there exists $c∈[0,1]$ such that $F(c)=0$, or $f(c)=c$, this is the fixed point we search for.