query: Prove that $f(x,y)$ is continuous on $\mathbb{R^2}$
positive_passages.docid: 155923
positive_passages.text: continuity of $f(x,y)$ in $\mathbb R^2$.
negative_passages.docid: 109896
negative_passages.text: Proving that the function $f(x,y)=\frac{x^2y}{x^2 + y^2}$ with $f(0,0)=0$ is continuous at $(0,0)$.
negative_passages.docid: 136011
negative_passages.text: Must $f$ be continuous?
negative_passages.docid: 15914
negative_passages.text: Prove that any polynomial function is continuous
negative_passages.docid: 149354
negative_passages.text: Proof of continuity $f(x+y) = f(x) + f(y)$
negative_passages.docid: 62048
negative_passages.text: Proving a Lipschitz function is continuous
negative_passages.docid: 153730
negative_passages.text: Show that $f(x)=x^2$ is continuous at $a=2$ using the $\delta-\epsilon$ definition of continuity.
negative_passages.docid: 17223
negative_passages.text: Continuity of an inverse function.
negative_passages.docid: 46834
negative_passages.text: show that a distance function is continuous
negative_passages.docid: 16735
negative_passages.text: If $f,g:X \to Y$ are continuous and $Y$ is $T_2$, then $\{x \in X\,|\,f(x)=g(x)\}$ is closed
negative_passages.docid: 97553
negative_passages.text: Continuity of $\frac{2xy}{x^2+y^2}$ at $(0,0)$
negative_passages.docid: 14578
negative_passages.text: Proving that an additive function $f$ is continuous if it is continuous at a single point
negative_passages.docid: 72694
negative_passages.text: Proving that $f(x) = g(x)$ for some $x \in [a,b]$ if $f,g$ continuous, $f(a)  g(b)$
negative_passages.docid: 55238
negative_passages.text: Show that if $f:X\to Y$ is a continuous function if and only if the graph of $f$ is a closed subset of $X\times Y$
negative_passages.docid: 149095
negative_passages.text: $f(\mathbb{R}\setminus \mathbb{Q}) \subseteq \mathbb{Q}$ and $f(\mathbb{Q}) \subseteq \mathbb{R}\setminus \mathbb{Q}$ imply that $f$ is not continuous
negative_passages.docid: 80643
negative_passages.text: Show that f(x) defined on [0,infinity) R to is continuous at 0.
negative_passages.docid: 58597
negative_passages.text: A characterization of functions from $\mathbb R^n$ to $\mathbb R^m$ which are continuous