Let $f:(a,b)\rightarrow \mathbb{R}$ be a continuous function. Suppose $\exists k\in (0,1)$ such that $\forall x,y\in (a,b), f(kx+(1-k)y)≦kf(x)+(1-k)f(y)$.
Let $A=\{\lambda\in [0,1]|\forall x,y\in (a,b) , f(\lambda x + (1-\lambda)y)≦\lambda f(x) + (1-\lambda)f(y)\}$.
Then $A$ is dense in $[0,1]$.
I have proved that $f$ is convex when $k$ is a rational, but what if $k$ is irrational? (in ZF)
I constructed a sequence in $A$ which is convergent to some fixed $p$ in $(0,1)$ when $k\in \mathbb{Q}$, but there must be a better proof using the definition of $\epsilon-\delta$.