Let $f:\mathbb R\to \mathbb R$ be coninuous. Suppose there exists $x_0$ such that $f(f(x_0))=x_0$. Prove that $f$ has a fixed point or in other words: $\exists c\in\mathbb R: f(c)=c$ .
Suppose $f(x_0)\neq x_0$ and there's some $x_1$ such that $f(x_1)=x_0$.
Then: $f(f(x_1))=f(x_0)\neq x_0$ and, $f(f(x_0))=f(x_2)=x_0$ but $f(x_1)=x_0$ and since the function is continuous there can't be $x_1\neq x_2: f(x_1)=f(x_2)$ so there's a contradiction.