Exercise 11, § 4

Let $f$ be a continuous function on an open interval $I \subset \mathbf{R}$, having a finite right derivative at every point of $I$. If for every $x \in I$ and every $y \in I$ such that $y > x$ the point $M_y = (y, f(y))$ lies above the right semi-tangent to the graph of $f$ at the point $M_x = (x, f(x))$, show that $f$ is convex on $I$ (using the mean value theorem show that

Give an example of a function which is not convex, has a finite right derivative everywhere, and such that for every $x \in I$ there exists a number $h_x > 0$ depending on $x$ such that $M_y$ lies above the right semi-tangent at the point $M_x$ for all $y$ such that $x \leq y \leq x + h_x$. This last condition is nevertheless sufficient for $f$ to be convex, if one supposes further that $f$ is differentiable on $I$ (use I, p. 12, corollary).