Superadditive but not convex function

Question

I want to analyze if there exists a continuous, superadditive, but not convex function, $f: [0,1] \to [0,1]$, such that $f(x+y)+f(1-y) \leq f(1)+f(x)$ for all $x,y \in [0,1]$ with $x+y \leq 1$.

Answer 1

We can assume that $x+y\leq 1-y$. Then consider $f(x)= \frac{3}{2}x,\ 0\leq x\leq \frac{1}{3} ,\ f(x)=\frac{1}{2},\ \frac{1}{3} <x<\frac{2}{3}$ and $f(x)=\frac{3}{2}x-\frac{1}{2},\ \frac{2}{3}\leq x\leq 1 $.