Exercise 11, § 4

Soit $f$ a continuous function in an open interval $I \subset \mathbf{R}$, admitting at every point of $I$ a finite right derivative. If, for all $x \in I$ and all $y \in I$ such that $y > x$, the point $M_y = (y, f(y))$ is above the right half-tangent at the point $M_x = (x, f(x))$ of the graph of $f$, now prove that $f$ is convex in $I$ (using the theorem of finite increments, show that one has $f'_d(y) \geqslant \frac{f(y) - f(x)}{y-x}$ for $x < y$).

Give an example of a non-convex function, having at every point a finite right derivative, and such that for every $x \in I$, there exists a number $h_x > 0$ depending on $x$, such that $M_y$ is above the right half-tangent at the point $M_x$ for every $y$ such that $x \leqslant y \leqslant x + h_x$. This last condition is nevertheless sufficient for $f$ to be convex, if one supposes in addition that $f$ is differentiable in $I$ (use I, p. 20, corollary).