Check the stated properties of the functions \(f\), \(g\), and \(k\) of Example 1 of §49.
Solution
Write \(I=[0,1]\). For \(0<h\le 1/2\), let \(\Delta f(x,h)\) be the larger, in absolute value, of the two secant quotients of step \(h\) that are defined at \(x\).
For \[ f(x)=4\alpha x(1-x) \] and \(h=1/4\), the forward quotient is \[ \frac{f(x+h)-f(x)}h=4\alpha(1-2x-h) =\alpha(3-8x), \] while the backward quotient is \[ \frac{f(x-h)-f(x)}{-h}=4\alpha(1-2x+h) =\alpha(5-8x). \] Whenever both are defined, their difference is \(2\alpha\), so at least one has absolute value at least \(\alpha\). Near an endpoint only one quotient may be defined; if \(0\le x<1/4\), then \(|3-8x|\ge1\), and if \(3/4<x\le1\), then \(|5-8x|\ge1\). Hence \[ \Delta f(x,1/4)\ge\alpha\qquad(x\in I). \] In particular, if \(\alpha>4\), then \(f\in U_4\).
The function \(g\) in Figure 49.1 is the tent function \[ g(x)= \begin{cases} \alpha x,&0\le x\le1/2,\\ \alpha(1-x),&1/2\le x\le1. \end{cases} \] Every linear piece has slope of absolute value \(\alpha\). Fix \(0<h\le1/4\) and \(x\in I\). If \(x\le1/2\), then either \(x+h\le1/2\), so the forward quotient has absolute value \(\alpha\), or else \(x\ge1/2-h\), in which case \(x-h\ge0\) and the backward interval \([x-h,x]\) lies in the left linear piece, so the backward quotient has absolute value \(\alpha\). The case \(x\ge1/2\) is symmetric. Therefore \[ \Delta g(x,h)\ge\alpha \] for every \(x\) and every \(0<h\le1/4\). Thus, if \(\alpha>n\) and \(1/n\le1/4\), choosing any \(0<h\le1/n\) gives \(g\in U_n\).
The function \(k\) is the two-tooth sawtooth obtained by joining successively \[ (0,0),\quad (1/4,\alpha/4),\quad(1/2,0),\quad (3/4,\alpha/4),\quad(1,0). \] Again every linear piece has slope \(\pm\alpha\). If \(0<h\le1/8\), then for every \(x\in I\), at least one of the two intervals \([x-h,x]\) or \([x,x+h]\) that is defined is contained in one of the four linear pieces: the vertices are spaced by \(1/4\), while \(h\le1/8\). Along that interval the corresponding secant quotient has absolute value \(\alpha\). Hence \[ \Delta k(x,h)\ge\alpha \qquad(0<h\le1/8). \] Consequently \(k\in U_n\) whenever \(\alpha>n\) and \(n\ge8\) (take, for example, \(h=1/n\)).
These are exactly the properties asserted for \(f,g,k\) in Example 1.