Exercise 1, § 5

Let I be a semi-open interval $]a, b]$ in $\mathbf{R}$, and let $F = \mathbf{R}^I$ be the space of all mappings of I into $\mathbf{R}$, equipped with the topology of pointwise convergence; for every $x \in I$, denote by $f(x)$ the mapping $t \mapsto |x - t|$ of I into $\mathbf{R}$, which is an element of F; the mapping f of I into F is continuous. Show that f is left-differentiable at every point of I, but the left-derivative $f'_l$ is a function (with values in F) that is not measurable for Lebesgue measure, even though it is the pointwise limit of a sequence of continuous functions and, for each $t \in I$, the function $\operatorname{pr}_t \circ f'_l$ is a measurable numerical function (note that $f'_l$ is not right-continuous at any point of I, and make use of Exer. 1 of GT, IV, §2).