Let $$D^+u(x) = \left\{v \in \mathbb{R}; \limsup_{y\to x}\frac{u(y)-u(x)-v\cdot (y-x)}{|y-x|} \le0\right\}$$ and
$$D^-u(x) = \left\{v \in \mathbb{R}; \liminf_{y\to x}\frac{u(y)-u(x)-v\cdot (y-x)}{|y-x|} \ge0\right\}$$ the superdifferential and the subdifferential of the function $u :O \to \mathbb{R}$ continuous with $O \subset \mathbb{R}$.
I am looking for a continuous function $u$ such that $D^+u(x)$ contains a single point and $D^-u(x)$ is empty for some point $x\in O$.
I know such example can not be semiconcave.