Proving differentiability for inverse function on given interval

[TanWu]

Homework Statement

Prove that $e:(c(-d), c(d)) \rightarrow(-d, d)$ is differentiable by computing $e^{\prime}(s)$ from the definition for each $s \in (c(-d), c(d))$. You may use the fact $\frac{2}{3} < \frac{e(t) - e(s)}{t - s} < 2$ for $t \neq s \in (c(-d), c(d))$ without proof.

Relevant Equations

$\lim_{t \to s} \frac{2}{3} < \lim_{t \to s} \frac{e(t) - e(s)}{t - s} < \lim_{t \to s} 2$

I am trying to solve (a) and (b) of this question.

(a) Attempt

We know that $\frac{2}{3} < \frac{e(t) - e(s)}{t - s} < 2$ for $t \neq s \in (c(-d), c(d))$

Thus, taking the limits of both sides, then

$\lim_{t \to s} \frac{2}{3} < \lim_{t \to s} \frac{e(t) - e(s)}{t - s} < \lim_{t \to s} 2$

Or equivalently,

$\frac{2}{3} < \lim_{t \to s} \frac{e(t) - e(s)}{t - s} < 2$

$\frac{2}{3} < e'(s) < 2$

Thus, using the squeeze principle, then $e'(s)$ is bounded between $\frac{2}{3}$ and $2$, then the derivative exists for $t \in (c(-d), c(d))$ and thus we have proved that $e(s)$ is differentiable on the required interval. Are we allowed to do that?

(b) Attempt

Since $c(x)$ is $C^1$ on (-d, d) and as we proved in (a), then e must be $C^1$ on $(e(-d), e(d))$ as it is differentiable and thus continuous. This seems somewhat too trivial of a question to me.

I express gratitude to those who help.

 

[Office_Shredder]

The squeeze principle requires the smallest and largest things to become equal in the limit. 2/3 and 2 are not equal, no matter how close t and s are...

 

[TanWu]

The squeeze principle requires the smallest and largest things to become equal in the limit. 2/3 and 2 are not equal, no matter how close t and s are...

Thank you Sir. I apologize over my mistake. Is my proof correct thought without reference to the squeeze principle thought? Like using the fact the derivative is bounded? I cannot think of any other ways of proving it.

 

[Office_Shredder]

I don't really understand what the point of letting you use that inequality is (especially since you proved it in question 7, so it's not like they're doing you a great favor). I honestly do not know what they expect the answer to look like. I assume you're supposed to take that ratio and express it in terms of c and then compute something that looks similar to the derivative of c.