TMA4225: Exercises for week 37

Show that \(f(x)=x^2 \sin(x^{-1})\) has bounded variation on \((0,1]\), while \(f(x)=x \sin(x^{-1})\) does not.

Prove the following change-of-variable formula for the Riemann–Stieltjes integral: Assume that \(f\) is Riemann–Stieltjes integrable with respect to \(g\) on the interval \([a,b]\), and that \(h\colon [c,d]\to[a,b]\) is a strictly increasing, continuous function with \(h(c)=a\), \(h(d)=b\). Then \(f\circ h\) is Riemann–Stieltjes integrable with respect to \(g\circ h\), and \[ \int_c^d f(h(t))\,d(g\circ h)(t)=\int_a^b f(x)\,dg(x).\] (Notation: \(g\circ h\) is the composition of \(g\) and \(h\), defined by \((g\circ h)(t)=g(h(t))\).

Later on, we shall study the Cantor function \(C\) in some detail. For this problem, assume the following facts as given: \(C\colon[0,1]\to[0,1]\) is non-decreasing and continuous, and \[ C(x)=\begin{cases} \textstyle\frac12C(3x)&0\le x\le\textstyle\frac13,\\ \textstyle\frac12&\textstyle\frac13\le x\le\textstyle\frac23,\\ \textstyle\frac12\bigl(1+C(3x-2)\bigr)&\textstyle\frac23\le x\le1. \end{cases} \] What is the value of \[\int_0^1 C(x)\,dC(x) ? \]

111X(c–d) and 111Y(c-e)