Problem 1
Prove that a function \(f\) is absolutely continuous on \([a, b]\) if and only if it is a continuous function of bounded variation mapping every subset \(Z \subset[a, b]\) of measure zero into a set of measure zero. G Generalizing Problem 9, p. 327 , by a jump function, we now mean a function of the form where the numbers \(h_{1}, \ldots, h_{\mathrm{n}}, \ldots\) and \(h_{1}, \ldots, h_{n}^{\prime}, \ldots\) corresponding to the discontinuity points \(x, \ldots, x_{n}, \ldots\) and \(x_{1}^{\prime} \ldots, x_{n}^{r} \ldots\) satisfy the conditions \(\sum_{n}\left|h_{\mathrm{s}}\right|<\infty, \quad \sum_{n}\left|h_{\mathrm{s}}^{\prime}\right|<\infty\) (we now allow negative \(\left.h_{n} h_{n}^{t}\right)\).
Problem 5
Let \(A_{[a, b]}^{0}\) be the space of all absolutely continuous functions f defined on \([a, b]\), satisfying the condition \(f(a)=0\). Prove that \(A_{[a, b]}^{0}\) is a closed subspace of the space \(V_{[a, b]}^{0}\) of all functions of bounded variation on \([\mathrm{a}, b]\) satisfying the same condition, equipped with the norm \(\|\mathbf{f}\|=V_{a}^{b}(f)\). Comment. There is no need for the condition \(f(a)=0\) if we regard all functions differing by a constant as equivalent. We then have \(\|\mathbf{f}\|=0\) if and only iff \(=\) const.
