query: Prove $f^{-1}$ is smooth
positive_passages.docid: 152326
positive_passages.text: What is the inverse of the function $x \mapsto \frac{ax}{\sqrt{a^2 - |x|^2}}$?
negative_passages.docid: 23071
negative_passages.text: Showing that $f_0 (x_1, \ldots, x_m) \mathrm tr A =\displaystyle{ \sum_{i=1}^n} f_0(x_1, \ldots, Ax_i,\ldots, x_m)$
negative_passages.docid: 55018
negative_passages.text: Prove that $A^*A + I$ is invertible
negative_passages.docid: 108593
negative_passages.text: First order weak derivatives of $f(x)=|x|^r$
negative_passages.docid: 98344
negative_passages.text: Prove that $\int_0^1[f''(x)]^2dx\ge4.$
negative_passages.docid: 80650
negative_passages.text: Complex differentiability of $f(z)=|z|$
negative_passages.docid: 33969
negative_passages.text: "Basis extension theorem" for local smooth vector fields
negative_passages.docid: 30860
negative_passages.text: Prove that $f(x)=0$ in $[-1,1]$.
negative_passages.docid: 14830
negative_passages.text: A transformation of F[x]
negative_passages.docid: 65923
negative_passages.text: How to approach proving $f^{-1}(B\setminus C)=A\setminus f^{-1}(C)$?
negative_passages.docid: 45432
negative_passages.text: Showing that $f:(0,2\pi]\to S^1$ is not a homeomorphism
negative_passages.docid: 124861
negative_passages.text: The extension of smooth function
negative_passages.docid: 33734
negative_passages.text: Every closed subset $E\subseteq \mathbb{R}^n$ is the zero point set of a smooth function
negative_passages.docid: 13899
negative_passages.text: Proof that $-v = (-1)*v$
negative_passages.docid: 134382
negative_passages.text: Proving bounded first derivative
negative_passages.docid: 104263
negative_passages.text: How to show that $f'(x)<2f(x)$
negative_passages.docid: 150274
negative_passages.text: Inverse of a function - $f^-(x)$ in a restricted domain