Suppose that $f$ is a one-to-one function and that $f^{−1}$ has a derivative which is nowhere $0$. Prove that $f$ is differentiable.
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1$\begingroup$ As a side-note, the cubic root on $\mathbb R$ is a counterexample if the derivative of $f^{-1}$ is zero somewhere. Its inverse is $x^3$ which is differentiable everywhere, but the cubic root isn't. $\endgroup$– Najib IdrissiJun 16, 2013 at 7:54
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$\begingroup$ This question is phrased as an imperative sentence ("Prove that") rather than as a question. Many consider questions phrased in the imperative to be impolite. Moreover, you have left out two key things in the question: where did you encounter it (what book, what course?) and more importantly: what have you already tried? $\endgroup$– Carl MummertJun 16, 2013 at 16:04
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$\begingroup$ Thanks.I am bad at english. In my first course on Calculus.Can you pls frame it correctly for me? $\endgroup$– RustyJun 16, 2013 at 16:35
1 Answer
Since $f$ is one-to-one, $f^{−1} $ is a function whose inverse is $f$.
Since $f^{−1} $ is differentiable, $f^{−1} $ is continuous.
Let $y = f(x)$ , so $f^{−1} (y) = x$.
$$\lim_{h\to0}\frac{f(x+h)−f(x)}{h}=\lim_{h\to0}\frac{f(x+h)−y}{h}.$$
Now since $f^{−1}$ is continuous, we can write $x + h = f^{−1} (y + k)$ where $k \to 0 $ as $ h \to 0.$
Now
$$\lim_ {h\to0} \frac {f(x+h)−y}{h} = \lim_{h\to0} \frac {f(f^{−1} (y+k))−y}{(x+h)−x} = \lim_{h\to0} \frac {(y+k)−y}{f^{−1} (y+k)−f^{−1} (y)} = \lim _{k\to0} \frac {k}{f^{−1} (y+k)−f^{−1} (y)} $$
as "$h \to 0 \implies k \to 0$".
Hence
$$ \lim_{h\to0} \frac {f(x+h)−y}{h} = \lim _ {k\to0}\frac {k} { f^{−1}(y+k) − f^{−1}(y) } = \frac {1} {(f^{−1})′(y) } = \frac {1} {(f^{−1})′(f(x)) }.$$
So f is differentiable.