Exercise 1, § 1

Let $(f_\alpha)$ be a set of real functions, defined on an interval $I \subset \mathbf{R}$, each admitting a strict primitive on $I$, and forming a directed set for the relation $\leq$. Let $f$ be the upper envelope of the family $(f_\alpha)$; suppose that $f$ admits a strict primitive on $I$. Show that if $g_\alpha$ (resp. $g$) is the primitive of $f_\alpha$ (resp. $f$) which vanishes at a point $x_0 \in I$, then $g$ is the upper envelope of the family $(g_\alpha)$ on the intersection $I \cap [x_0, +\infty[$ and its lower envelope on the intersection $I \cap ]-\infty, x_0]$. (Restricting to the first of these intervals, show that if $u$ is the upper envelope of $(g_\alpha)$ on $I \cap [x_0, +\infty[$ then one has $u(x+h) - u(x) \leq g(x+h) - g(x)$ for $h > 0$; then prove that, for all $\alpha$, one has $u(x+h) - u(x) \geq g_\alpha(x+h) - g_\alpha(x)$, and conclude from this last inequality that $\liminf_{h \to 0^+} (u(x+h) - u(x))/h \geq f(x)$; deduce the proposition from this.)

Give an example of an increasing sequence $(f_n)$ of continuous functions on an interval $I$ which are uniformly bounded, but whose upper envelope does not admit a strict primitive on $I$.