Real Analysis: Problems with Solutions

Solution:
Consider the function [tex]g(x)=f(x)-x[/tex] on [tex][0, 1][/tex]; it is continuous as a difference of continuous functions. Since the values of [tex]f[/tex] lie in [tex][0, 1][/tex],
[tex]g(0)=f(0)\ge 0,\qquad g(1)=f(1)-1\le 0[/tex]
If [tex]g(0)=0[/tex], then [tex]c=0[/tex] is a fixed point; if [tex]g(1)=0[/tex], then [tex]c=1[/tex] is one. Otherwise [tex]g(0)\gt 0\gt g(1)[/tex], and by the intermediate value theorem (Bolzano's theorem) the continuous function [tex]g[/tex] vanishes at some point [tex]c\in(0, 1)[/tex]:
[tex]g(c)=0\iff f(c)=c[/tex]