Constructing a continuous map for an interval with some points of period x, but none of period 3

Question

I was given this question where I have to construct a continuous map f:I -> I (interval) with a point of period 4, but none of period 3. I know that thanks to Sharkovskii's theorem that if it had a point of period 3, it would have a point of any period implying chaos, so this map I need to construct must not be chaotic. I know the function must be such that

$(f)^4$(x) = x, ie => $(f)^4$(x) - x = 0

The only problem is I don't know how to go about constructing a map, any starting tips would be greatly appreciated.

Answer 1

Compilation of comments. Define $f(x)=x+1/4 \pmod 1$. It is a continuous function $I\to I$. $f(0)=1/4, f^2(0)=1/2, f^3(0)=3/4, f^4(0)=0$, so $0$ is a periodic point of period $4$. It is easy to see that there are no periodic points of period $3$.