91Ó°ÊÓ

Let \(f:[0,1] \rightarrow \mathbb{R}\) be defined by \(f(0)=0\) and \(f(x)=x^{2} \sin \left(\pi / x^{2}\right)\) for \(x \in(0,1] .\) Given any \(n \in \mathbb{N}\), consider the partition $$ P_{n}:=\left\\{0, n^{-1 / 2},\left(n-\frac{1}{2}\right)^{-1 / 2},(n-1)^{-1 / 2}, \ldots,(3 / 2)^{-1 / 2}, 1\right\\} $$ of \([0,1]\) and write \(P_{n}:=\left\\{x_{0}, x_{1}, \ldots, x_{2 n-2}\right\\} .\) Show that $$ \sum_{i=1}^{2 n-2} \sqrt{\left(x_{i}-x_{i-1}\right)^{2}+\left(f\left(x_{i}\right)-f\left(x_{i-1}\right)\right)^{2}} \geq\left(\frac{1}{3}+\frac{1}{5}+\cdots+\frac{1}{2 n-1}\right) $$ Deduce that the curve \(y=f(x), 0 \leq x \leq 1\), is not rectifiable even though the function \(f\) is differentiable. (Hint: Exercise 2.10.)

Let \(f:[a, b] \rightarrow \mathbb{R}\) be a bounded function that is continuous on \((a, b)\), and let \(w:[a, b] \rightarrow \mathbb{R}\) be a weight function that is continuous and positive on \((a, b) .\) Show that there exists \(c \in(a, b)\) such that \(\operatorname{Av}(f ; w)=f(c)\). (Hint: Apply the Cauchy Mean Value Theorem (Proposition \(4.38\) ) to the functions \(F, G:[a, b] \rightarrow \mathbb{R}\) defined by \(F(x):=\int_{a}^{x} f(t) w(t) d t\) and \(G(x):=\) \(\left.\int_{a}^{x} w(t) d t .\right)\)

Let \(f:[a, b] \rightarrow \mathbb{R}\) be a bounded function that is continuous on \((a, b)\). If the range of \(f\) is contained in \((\alpha, \beta)\) and \(\phi:[\alpha, \beta] \rightarrow \mathbb{R}\) is a convex function that is continuous at \(\alpha\) and \(\beta\), then show that \(\operatorname{Av}(f) \in(\alpha, \beta)\), the function \(\phi \circ f:[a, b] \rightarrow \mathbb{R}\) is integrable, and \(\phi(\mathrm{Av}(f)) \leq \mathrm{Av}(\phi \circ f)\).