Let $f(x)$ be a continuous function from $\mathbb{R}\rightarrow\mathbb{R}$.
Let's denote $k$-times repeated application of the function, $f(f(f(...f(x)...)))$ as $f^{(k)}(x)$.
Let's call any $x$ a periodic point with period $n$ if $f^{(n)}(x)=x$.
Is it true that if a point with period 3 exists, then points with all possible periods exist?
In other words is it true that
$$\exists x:f^{(3)}(x)=x\Rightarrow \forall n>0 \exists y:f^{(n)}(y)=y$$
and if so, why?