Exercise 19, § 4

Soient $f$ et $g$ two finite numerical functions, $f$ being defined and continuous in an interval $I$, $g$ defined and continuous in $\mathbf{R}$. Suppose that, for every pair $(\lambda, \mu)$ of real numbers, $g(f(x)) + \lambda x + \mu$ is convex in $I$. (a) Now prove that $g$ is convex and monotone in $\mathbf{R}$. (b) If $g$ is increasing (resp. decreasing) in $\mathbf{R}$, now prove that $f$ is convex (resp. concave) in $I$ (use Prop. 7).