query: Show that there is a function $g: \mathbb{R}^{2} \to \mathbb{R}$ of class $C^{\infty}$ such that $f(x,y) = g(x,y)\cdot x\cdot y$
positive_passages.docid: 23186
positive_passages.text: $f: \mathbb{R^2} \to \mathbb{R}$ a $C^\infty$ function such that $f(x,0)=f(0,y)=0$ then exists $g$ such that $f(x,y)=xy\, g(x,y)$
negative_passages.docid: 11453
negative_passages.text: Find a real function $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x)) = -x$?
negative_passages.docid: 83036
negative_passages.text: f(g(x)) has a degree divisible by n
negative_passages.docid: 153612
negative_passages.text: Finding $g_i:\mathbb{R}^n\to\mathbb R$ s.t $f(x)=\sum\limits_{i=1}^nx_i\cdot g_i(x)$
negative_passages.docid: 109610
negative_passages.text: Is $\mathbb{R}[X]/(P)$ isomorphic to $\mathbb C$ for every irreducible polynomial $P$ of degree $2?$
negative_passages.docid: 146480
negative_passages.text: Example of $\deg(fg)<\deg(f)+\deg(g)$
negative_passages.docid: 134043
negative_passages.text: Prove that for $ f(x+2\cdot\pi)=f(x), x \in \mathbb{R} $, there exists $ x_0 $ so $ f(x_0+\pi)=f(x_0) $
negative_passages.docid: 34260
negative_passages.text: Injective function from $\mathbb{R}^2$ to $\mathbb{R}$?
negative_passages.docid: 105319
negative_passages.text: A calculus problem with functions such that $f''(x) = g(x)$ and $g''(x) = f(x)$
negative_passages.docid: 58597
negative_passages.text: A characterization of functions from $\mathbb R^n$ to $\mathbb R^m$ which are continuous
negative_passages.docid: 125788
negative_passages.text: A function with a property $f(x+y)=f(x)f(y)$
negative_passages.docid: 124562
negative_passages.text: $f:\mathbb{R}^n \to \mathbb{R}$ has expansion $\sum_i g_i(x)x^i$
negative_passages.docid: 138931
negative_passages.text: Show that $||x+y||\cdot||x-y||\le||x||^2+||y||^2$
negative_passages.docid: 25010
negative_passages.text: Argue that there exists a continuous function $g: \mathbb{R} \rightarrow \mathbb{R}$ which satisfies $g(x) = \frac{1-e^{-2x^2}}{x^2}$ for $x \neq 0$
negative_passages.docid: 118154
negative_passages.text: $f: \mathbb{R} \to \mathbb{R}$ that takes each value in $\mathbb{R}$ twice
negative_passages.docid: 104263
negative_passages.text: How to show that $f'(x)<2f(x)$
negative_passages.docid: 39090
negative_passages.text: Finding functions such that $f(g(x)) = f(x)g(x)$