Chapter 32

Prove that the function $$ f(x)= \begin{cases}x^{\alpha} \sin \frac{1}{x^{\beta}} & \text { if } \quad 0\beta\) but not if \(\alpha \leqslant \beta\).

Suppose \(\mathbf{f}\) has a bounded derivative on \([\mathrm{a}, \mathrm{b}]\), so that \(f^{\prime}(x)\) exists and satisfies an inequality \(\left|f^{\prime}(x)\right|

Let \(\mathbf{f}\) be a function of bounded variation on \([a, b]\) such that $$ f(x)>c>0 $$ Prove that \(1 / f\) is also a function of bounded variation and $$ V_{a}^{o}\left(\frac{1}{f}\right)<\frac{1}{c^{2}} V_{a}^{b}(f) $$

Let \(\mathbf{f}\) be a function of bounded variation on \([\mathrm{a}, \mathrm{b}]\). Prove that $$ \|f\|=V_{a}^{b}(f) $$ has all the properties of a norm (cf. p. 138 ) if we impose the extra condition \(\mathbf{f}(\mathrm{a})=0\) Comment. Thus the space \(V_{[a, b]}^{0}\) of all functions of bounded variation on \([a, b]\) equipped with this norm and vanishing at \(x=a\) is a normed linear space (addition of functions and multiplication of functions by numbers being defined in the usual way).