query: Let $f $ be $C^{\infty}$ with $f(x,0)=f(0,y)=0\in\mathbb R$. Show $f(x,y) = xyg(x,y)$ for $g\in C^\infty$
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: 145233
negative_passages.text: How to show $\lim_{n \rightarrow \infty} \int_0^1 n x^n f(x) \; dx = f(1)$ for continuous $f$?
negative_passages.docid: 91616
negative_passages.text: Existence of a $C_c^{\infty}$ function $\Phi$ s.t. $\int \nabla \Phi \neq 0.$
negative_passages.docid: 79908
negative_passages.text: Show $\int_0^\infty \frac{e^{-x}-e^{-xt}}{x}dx = \ln(t),$ for $t \gt 0$
negative_passages.docid: 158011
negative_passages.text: $C_c^{\infty}(\mathbb R^n)$ is dense in $W^{k,p}(\mathbb R^n)$
negative_passages.docid: 76689
negative_passages.text: How to prove $C_1 \|x\|_\infty \leq \|x\| \leq C_2 \|x\|_\infty$?
negative_passages.docid: 20467
negative_passages.text: Show that $(y_n)\in l^q$
negative_passages.docid: 10747
negative_passages.text: Given function $g$ continuous in line $[0,\infty)$...
negative_passages.docid: 88690
negative_passages.text: $\int_a^b f(x)\mathrm dx=0$ for all $-\infty<a<b<\infty$ a,b rational implies $f=0$ a.e
negative_passages.docid: 126461
negative_passages.text: Show that $g:I\to\mathbb{R}$ is continuous and $f:(x-x_0)g(x)$ is differentiable
negative_passages.docid: 154117
negative_passages.text: Showing $[(fg)h]_n = [f(gh)_n]$ in the linear algebra $F^\infty$
negative_passages.docid: 11828
negative_passages.text: continuous functions on $\mathbb R$ such that $g(x+y)=g(x)g(y)$
negative_passages.docid: 28949
negative_passages.text: Prove $f(x) = 0 $for all $x \in [0, \infty)$ when $|f'(x)| \leq |f(x)|$
negative_passages.docid: 22303
negative_passages.text: How do I prove that $f(x)f(y)=f(x+y)$ implies that $f(x)=e^{cx}$, assuming f is continuous and not zero?
negative_passages.docid: 68409
negative_passages.text: Consider $f(x,y)=\sup_{θ,φ}\{||e^{i θ }x+ e^{i φ }y||^2: θ,φ∈R\}$ in which $x,y\in\mathbb C^n$
negative_passages.docid: 40564
negative_passages.text: Prove that $u(x,t)=\int_{-\infty}^{\infty}c(w)e^{-iwx}e^{-kw^2t}dw\rightarrow 0$ if $x\rightarrow \infty$
negative_passages.docid: 42108
negative_passages.text: Prove that $c_0$ is closed in $\ell^\infty$