query: If $f\colon \mathbb{R} \to \mathbb{R}$ is one-to-one and differentiable at $a$ with $f'(a) \ne 0$, must $f^{-1}$ be differentiable at $f(a)$?
positive_passages.docid: 151575
positive_passages.text: Differentiable bijection $f:\mathbb{R} \to \mathbb{R}$ with nonzero derivative whose inverse is not differentiable
negative_passages.docid: 136714
negative_passages.text: Proving that $\lim_{x \to 0} \frac{f(x)}{|x|}=1$ implies $f$ is not differentiable at $0$.
negative_passages.docid: 111458
negative_passages.text: Prove that $f$ is differentiable
negative_passages.docid: 99827
negative_passages.text: For what values will f(x) be necessarily one-one?
negative_passages.docid: 849
negative_passages.text: $f$ differentiable, $f(x)$ rational if $x$ rational; $f(x)$ irrational if $x$ irrational. Is $f$ a linear function?
negative_passages.docid: 43874
negative_passages.text: Differentiability implies continuity - A question about the proof
negative_passages.docid: 115635
negative_passages.text: Differentiability of antiderivative and continuity of derivative
negative_passages.docid: 40209
negative_passages.text: $f : S^1 \to\mathbb R$ is continuous then $f(x)=f(-x)$ for some $x\in S^1$
negative_passages.docid: 124346
negative_passages.text: Where is $f(z)=\Re (z)$ differentiable?
negative_passages.docid: 72115
negative_passages.text: Why isn't there a continuously differentiable surjection from $I \to I \times I$?
negative_passages.docid: 100765
negative_passages.text: If $f'(x) = 0$ for all $x \in \mathbb{Q}$, is $f$ constant?
negative_passages.docid: 130322
negative_passages.text: Show that $f(x) = 0$ if $x = 0$ $f(x) = \text{exp}(-1/x^{2})$ if $x \neq 0$ is infinitely differentiable but not analytic at $x_{0} = 0$ `
negative_passages.docid: 85623
negative_passages.text: Proof of continuity implied by differentiability
negative_passages.docid: 13575
negative_passages.text: Need help for proving that: $f(f^{-1}(A)) ⊆ A$
negative_passages.docid: 61782
negative_passages.text: How to prove that $f(x) = \frac{-2x+1}{(2x-1)^2-1}$ is one-to-one on $(0,1)$?
negative_passages.docid: 142870
negative_passages.text: Prove the function is not differentiable at (0,0)
negative_passages.docid: 96973
negative_passages.text: If $F(0,0)=0$ and $F(x,y)= \frac{xy}{x^2+y^2}$ for $(x,y)\neq (0,0)$ then $F$ is differentiable at $(0,0)$?