Suppose that \(g\) is a positive real-valued function of a real variable, not necessarily continuous. If \(a<b\) are real numbers, show that there is a finite sequence \[ a=t_0<t_1<\cdots<t_n=b \] such that in each interval \([t_k,t_{k+1}]\) there is a point where the value of \(g\) is greater than the length of the interval.
Solution
Call a finite partition \[ a=t_0<t_1<\cdots<t_m=c \] of \([a,c]\) good if for every \(0\leq k<m\) there is a tag \[ \xi_k\in[t_k,t_{k+1}] \] such that \[ t_{k+1}-t_k<g(\xi_k). \] Let \[ E \coloneqq \{c\in[a,b]:[a,c]\text{ has a good partition}\}. \]
1. The set \(E\) is nonempty.
The degenerate interval \([a,a]\) has the empty partition, so \(a\in E\). Alternatively, since \(g(a)>0\), every sufficiently small \(c>a\) lies in \(E\) by using the one-interval partition \([a,c]\) tagged at \(a\).
2. Let\[ c\coloneqq\sup E. \]Then\[ c>a. \]
Since \(g(a)>0\) and \(b>a\), choose \[ 0<h<\min(g(a),b-a). \] The one-interval partition \[ [a,a+h] \] tagged at \(a\) is good because \[ h<g(a). \] Thus \(a+h\in E\), so \(\sup E>a\).
3. One has\[ c=b. \]
Suppose instead that \(c<b\). Since \(g(c)>0\), choose \[ 0<\eta < \min\left( \frac{b-c}{2}, \frac{g(c)}{4} \right). \] By the definition of supremum, there is some \[ x\in E \] with \[ c-\eta<x\leq c. \] Set \[ y=c+\eta. \] Then \[ y<b \] and \[ y-x < 2\eta < g(c). \] Moreover \[ c\in[x,y]. \]
Take a good partition of \([a,x]\) and append the interval \([x,y]\) with tag \(c\). The appended interval is good because its length is less than \(g(c)\). Thus \[ y\in E, \] contradicting \(y>c=\sup E\). Therefore \(c=b\).
4. There is some\[ x\in E \]such that\[ b-x<g(b). \]
By step 3, \[ \sup E=b. \] Since \(g(b)>0\), choose a point of \(E\) in the interval \[ \left( b-\min(g(b),b-a), b \right]. \] For such an \(x\), the displayed inequality holds. If the chosen point is \(b\), the conclusion is already immediate.
5. The interval \([a,b]\) has a good partition.
4. If \(x=b\), this is true because \(b\in E\).Otherwise, take a good partition of \([a,x]\) and append the interval\[ [x,b] \]tagged at \(b\). Its length satisfies\[ b-x<g(b), \]so the enlarged partition is good.
6. Therefore there is a finite sequence\[ \boxed{ a=t_0<t_1<\cdots<t_n=b } \]such that every interval \([t_k,t_{k+1}]\) contains a point \(\xi_k\) with\[ g(\xi_k)>t_{k+1}-t_k. \]
Unpack the definition of the good partition furnished by step 5.
7. Q.E.D.
Step 6 is exactly the required statement.