Analyzing the convexity of a function

Question

If $f: [0,1] \to [0,1]$ is an increasing and continuous function with $f(0)=0$, $f(1)=1$, and $$f(x)+f(1-x) \leq 1$$ for all $x \in [0,1]$, I want to analyze if $f$ is convex. I think that the result is true. My idea is to show that $f$ is mid-point continuous, which implies that $f$ is convex, given the continuity of $f$.

Answer 1

Consider

$$f(x) = 4(x-1/2)^3 + \frac{1}{2}.$$