query: Prove every continuous $f$ can be written as $g-h$ where $g,h$ are continuous, non-negative functions
positive_passages.docid: 20091
positive_passages.text: Every continuous function is the difference of two nonnegative continuous functions
negative_passages.docid: 131992
negative_passages.text: A problem on continuous functions
negative_passages.docid: 118207
negative_passages.text: Prove that the product of two continuous functions is continuous
negative_passages.docid: 15914
negative_passages.text: Prove that any polynomial function is continuous
negative_passages.docid: 130299
negative_passages.text: Prove that if a function $f: X\to Y$ continuous then its graph is closed
negative_passages.docid: 57180
negative_passages.text: Techniques to prove a function is uniformly continuous
negative_passages.docid: 125682
negative_passages.text: Proof verification: the product of two continuous functions is continuous
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: 136011
negative_passages.text: Must $f$ be continuous?
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: 40297
negative_passages.text: Proof of uniform limit of Continuous Functions
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: 58597
negative_passages.text: A characterization of functions from $\mathbb R^n$ to $\mathbb R^m$ which are continuous
negative_passages.docid: 52922
negative_passages.text: let $f \colon [a,b] \to \mathbb{R}$ be continuous on $[a,b]$ and..
negative_passages.docid: 93021
negative_passages.text: $f:X\longrightarrow\mathbb{R}$ is continuous iff $\{x\in X:f(x)\geq\alpha\}$ and $\{x\in X:f(x)\leq\alpha\}$ are closed $\forall\alpha\in\mathbb{R}$
negative_passages.docid: 97104
negative_passages.text: Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ x$
negative_passages.docid: 100740
negative_passages.text: proof of that a continuous function has a fixed point