Any hint as to how to go about solving this would be helpful.
Consider a discrete-time martingale $x_n \in [0,1]$, i.e, $\mathbb{E}(x_{n+1}| \mathcal{F}_n) = x_{n}$, where $\mathcal{F}$ is the sigma-algebra.
Consider a function $f$ defined on $[0,1]$ such that, for some $0 $ f = \left\{ \begin{array}{ll}
f_1 & \mbox{on $[0,a]$};\\
f_2 & \mbox{on $[a,1]$}.\end{array} \right. $ where $f_1$ and $f_2$ are distinct concave functions. What can we say about $f(x_n)$? Is it a super-martingale? What if there is a discontinuity at $a$, i.e, $~f_1(a) \neq f_2(a)$? What can we say about $f(x_n)$ then? If $a=1$ or $a=0$, then it is easy to prove as a concave function of a martingale is a super-martingale.