I have several questions regarding the following problem.
Calculus by Michael Spivak, Chapter 13, Problem 21 (3rd Edition) begins as follows:
13-21. Suppose that $f$ is increasing. Figure 16 suggests that
$$\int_a ^b f^{-1} = bf^{-1}(b) - af^{-1}(a) - \int_{f^{-1}(a)} ^{f^{-1}(b)}f.$$
(a) If $P = \{t_0, \dots, t_n\}$ is a partition of $[a,b]$, let $P' = \{f^{-1} (t_0),\dots,f^{-1}(t_n)\}$. Prove that, as suggested in Figure 17,
$$L(f^{-1}, P) + U(f,P') = bf^{-1}(b) - af^{-1}(a).$$
(b) Now prove the formula stated above.
First, a few notes:
- Spivak means $f$ is strictly increasing. It's his custom to use "increasing" synonymously with "strictly increasing".)
- This chapter is where integrals are introduced. They are defined in terms of Upper and Lower Sums. Integration by parts, the Fundamental Theorem(s) of Calculus, Riemann sums, and other useful results are not yet available.
- The exercise just before this one gives a relevant result: that if a function $f$ (not necessarily continuous) is nondecreasing on $[c,d]$, $f$ is integrable on $[c,d]$. In our problem, $f$ is not assumed to be continuous. However, $f$ is increasing so it's integrable on closed intervals.
Spivak begins part (a) by picking some partition $P$ of $[a,b]$ and matching it up with a partition $P'$ of $[f^{-1}(a), f^{-1}(b)]$. Notice $P$ is not a partition on the domain of $f$: $a$ and $b$ are in the domain of $f^{-1}$.
Spivak seems to assume that for any point $t_i$ in $[a,b]$ there is a corresponding $t'_i$ in the domain of $f$ such that $f(t'_i) = t_i$. In other words, there are no gaps in the domain of $f^{-1}$.
This seems unjustified.
All we know from the hypothesis is that $f$ is increasing (and hence, so is $f^{-1}$) .
What if $f$ contains a jump discontinuity? For example, suppose $f$ is defined as
$$f(x) = \begin{cases} x, & \text{if $1\leq x< 2$} \\ x+1, & \text{if $x \geq 2.$} \end{cases} $$
This $f$ is increasing, as required. Consider $f$ on the interval $[1,3]$. This corresponds with $f^{-1}$ and the interval $[1,4]$.
If we now create a partition $P$ on $[1,4]$ such that $P$ includes the point $t_i = 2.5$, there is no corresponding $t'_i$ such that $f(t'_i) = 2.5$. $f^{-1}$ is only defined for points in $[1,2)$ or $[3,4]$.
It seems that a necessary requirement when selecting partition points of $P = \{a, t_1,\dots, t_{n-1},b\}$ as Spivak does, is that every $t_i$ must be in the domain of $f^{-1}$.
Is this right?
Until this problem, the book has dealt only with partitions of intervals on the domain of $f$. $f$ was always defined (and bound) over the intervals on which partitions were created.
But here we cannot arbitrarily partition $[a,b]$ even if $f$ is defined on the interval $[f^{-1}(a), f^{-1}(b)]$.
While trying to sort through this, I ended up beginning instead with a partition $P' = \{c, t_1, \dots,d\}$ where $[c,d]$ is in the domain of $f$. I then used $P'$ to generate the partition $P = \{f(c),f(t_1),\dots, f(d)\}$.
Using these partitions and otherwise following the approach of (a) results in an expression that's slightly different looking than (but equivalent to) the one desired: $$L(f^{-1}, P) + U(f,P') = f(d)d - f(c)c.$$
I then found, for the similar sum: $$U(f^{-1}, P) + L(f,P') = f(d)d - f(c)c.$$
Setting the left-hand sides of the above expressions equal to each other, we can arrive at $$U(f,P') - L(f,P') =U(f^{-1}, P)- L(f^{-1}, P)$$
Thus, integrability of $f$ on $[c,d]$ implies the integrability of $f^{-1}$ on $[f(c), f(d)]$.
I should note, The partition $P$ and the upper and lower sums for $f^{-1}$ basically ignore the fact that $f^{-1}$ may not be defined for all $x$ in $[f(c), f(d)]$. We're defining the integral $\int_{f(c)} ^{f(d)} f^{-1}$ in a way that will include some area contribution even from any "missing" sections.
Hence, If $f$ is increasing on $[c,d]$, $\int_{f(c)} ^{f(d)} f^{-1}$ exists, even if $f^{-1}$ is not defined for all points in $[f(c),f(d)]$.
Is this correct?
Is there a standard way of defining integrals over intervals on which the function may not be fully defined?
This seems like an important point, even if most of the rest of the book deals only with continuous functions.
The published solution for part (b) begins
It follows from (a) that $$ \int_a ^b f^{-1} = \sup\{L(f^{-1}, P)\} = \sup\{bf^{-1}(b) - af^{-1}(a) - U(f,P')\}$$
Here, Spivak uses the expression derived in (a) along with the definition of the integral. However, he seems to be assuming the integrability of $f^{-1}$, i.e. he never bothers to do the thing I did to show $U(f,P') - L(f,P') =U(f^{-1}, P)- L(f^{-1}, P)$. His
$$\sup\{L(f^{-1}, P)\} = \sup\{bf^{-1}(b) - af^{-1}(a) - U(f,P')\},$$ is true, but the expression $$\int_a ^b f^{-1} = \sup\{L(f^{-1}, P)\},$$ may not be.
I wonder if Spivak meant for us to use $f^{-1}$ nondecreasing to decide it's integrable? I don't think this result applies here, as it comes from functions that are known to be defined over the entire interval, and as we've seen, we cannot assume this is the case for our $f^{-1}$ here.
In other words, $f$ increasing and defined for all $[c,d]$ implies $f^{-1}$ is integrable on $[f(c),f(d)]$ (using this weird version of an integral that includes points on which $f^{-1}$ may not be defined), but this comes as a result of this problem. It would not have been correct to assume $f^{-1}$ is integrable only from $$\sup\{L(f^{-1}, P)\} = \sup\{bf^{-1}(b) - af^{-1}(a) - U(f,P')\}.$$
Anyway, it's interesting that the desired formula appears to work even if $f$ in not continuous, if we allow $\int f^{-1}$ to be defined in this weird way.
Summary of questions:
- When selecting partition points of $P = \{a, t_1,\dots, t_{n-1},b\}$ as Spivak does in part (a), must every $t_i$ be in the domain of $f^{-1}$?
- Suppose $f$ is increasing and is defined on $[c,d]$, but that $f$ has a jump discontinuity at a point $x_0$ in $(c,d)$. Is the integral $\int_{f(c)} ^{f(d)} f^{-1}$ defined?
- Is Spivak jumping the gun with his statement that $\int_a ^b f^{-1} = \sup\{L(f^{-1}, P)\}$, before first establishing the existence of $\int_a ^b f^{-1}$?