Is monotonicity a necessary condition for the inverse function theorem

Question

A textbook I was reading, Introduction To Real Analysis By Robert G. Bartle (page 169) states that the inverse theorem is defined as:

Let $I$ be an interval in $\mathbb{R}$ and let $f: I \rightarrow \mathbb{R}$ be strictly monotone and continuous on $I$. Let $J := f(I)$ and let $g: J \rightarrow \mathbb{R}$ be the strictly montone and continuous function inverse to $f$. If $f$ is differentiable at $c \in I$ and $f'(c) \neq 0$, then $g$ is differentiable at $d := f(c)$ and
$$
g'(d) = \frac{1}{f'(c)} = \frac{1}{f'(g(d))}
$$

However, in the proof given using Caratheodory's theorem in the book, the fact of the functions $f$ or $g$ being monotone was not used. Also, it seems a bit strange that only this should apply to only monotone functions. I can understand that if $f$ is monotone, then $g$ is monotone by continuous inverse theorem. But is this really necessary for the inverse function theorem to be used?

Answer 1

You must have a different edition of Bartle's text than I do (I have the 3rd edition, p. 164) - Bartle says here, in defining the inverse of a function:

If $f$ is a continuous strictly monotone function on an interval $I$, then its inverse function $g = f^{-1}$ is defined on the interval $J = f(I)$ and satisfies the relation $g(f(x)) = x$ for $x \in I$.

What this means, then, is that the inverse of $f$ may only be defined if $f$ is strictly monotone and continuous. You thus cannot claim the existence of the function $g$ in the claim you've provided above unless $f$ is strictly monotone and continuous.