Spivak's Calculus: Chapter 13 Question 21 I'm having a bit of trouble with the part b of the following question from Spivak's Calculus:

In particular, I'm not certain about what assumptions regarding the function $f^{-1}$ and $f$, I'm allowed to make, based on the information of the question. Can I assume that $f^{-1}$ is defined everywhere on [a, b], and bounded, and that f is integrable?
And if not, what can I assume about $f^{-1}$ and $f$? Thanks in advance! Could you also avoid giving hints on how to actually solve the question, as I would still like to attempt it myself.
 A: Here's a proof that an increasing function defined on $[a,b]$ is integrable. https://www.math.utah.edu/~yael/3210_public/exams/Integral.pdf .  Assuming increasing means strictly increasing,  it is also one to one so $f^{-1}$ is also defined, and is easy to be seen to be monotone as well, thus is integrable as well..  A monotone function on a closed interval is also trivially bounded, by $f(a)$ and $f(b)$
A: I believe this problem and Spivak's solution run into trouble if there are any jump discontinuities.
Though he doesn't say so, I suspect he may be implicitly (maybe even unknowingly) relying on $f$ being continuous on $(f^{-1}(a), f^{-1}(b))$. Without this condition, a jump discontinuity would cause a gap in the domain of $f^{-1}$.
The theorem still works in that case, sort of, if we allow $\int_a^bf^{-1}$ to include a chunk of area over the interval on which $f^{-1}$ isn't actually defined, but I doubt this was intended.
See here for more details (contains spoilers).
I'm not 100% certain on this, as my linked question never received any answers.
Update: In the case of a jump discontinuity, the formula can be applied to a function $g$, where $g$ takes on the value of $f^{-1}$ over intervals where it's defined, and takes on the constant value $x_0$ on the interval where $f^{-1}$ is not defined, where $x_0$ is the point at which $f$ "jumps".
I'll add some details to my linked question when I have some time.
