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Let $f$ a continuous and defined function on $\Bbb R$ such as: $$(\exists a \in \Bbb R) \quad f \circ f(a)=a$$ Prove that : $(\exists c \in \Bbb R) \quad f(c)=c$

As a try, I've used the definition of continuity and I wonder if I'm able to put $f(c)=a$ or $f(a)=c$ , any idea ?

Marios Gretsas
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1 Answers1

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There are three possibilities: (1) $f(a)=a$, (2) $f(a)<a$, (3) $f(a)>a$.

In case (1) we are done: take $c=a$.

In case (2) let $b=f(a)$, so $b<a$. We have $f(b)=a>b$. Now consider the function $g(x)=f(x)-x$. We have $g(b)=a-b>0$ and $g(a)=b-a<0$. But $f$ is continuous, so $g$ must be continuous, so $g(c)=0$ for some $b<c<a$.

A similar argument works for (3).

almagest
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