Prove a function with given properties has a fixed point

by RoRi

September 5, 2015

Given a real function $f$ continuous on $[a,b]$ , with $f(a) \leq a$ and $f(b) \geq b$ . Prove there is some $c \in [a,b]$ such that $f(c) = c$ (i.e., $f$ has a fixed point in $[a,b]$ ).

Proof. If $f(a) = a$ or $f(b) = b$ then we are done since these would be fixed points.

Assume then that $f(a) < a$ and $f(b) > b$ and let $g(x) = f(x) - x$ . We know that $g$ is continuous on $[a,b]$ and

$g(a) = f(a) - a < 0, \qquad \text{and} \qquad g(b) = f(b) - b > 0.$

Thus, by Bolzano’s theorem, there is some $c \in [a,b]$ such that $g(c) = 0$ ; hence, $f(c) - c = 0$ or $f(c) = c. \qquad \blacksquare$

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Prove a function with given properties has a fixed point

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