Exercise 19, § 1

Let f be a convex function in an interval I, h a number > 0, and $I_h$ the intersection of I and the intervals $I + h$ and $I - h$; now prove that, if $I_h$ is not empty, the function

is convex in $I_h$; if $h < k$, one has $g_h \leq g_k$. When h tends to 0, now prove that $g_h$ tends uniformly to f in every compact interval contained in the interior of I.