query: Let $g(x)$ be continuous for all $x$ and let $g(a+b) = g(a) + g(b)$ for all $a$,$b$...
positive_passages.docid: 13086
positive_passages.text: Overview of basic facts about Cauchy functional equation
negative_passages.docid: 138105
negative_passages.text: Let $f$ and $g$ be continuous at $ x=a$, with $f(a)<g(a)$. Prove that there exists a neighborhood $I$ of a such that $f(x)<g(x)$ for all $x \in I$.
negative_passages.docid: 144614
negative_passages.text: If $f$ is continuous on $[a,b]$ and $F(x) = \sup f([a,x])$. Prove that $F$ is continuous on $[a,b]$ .
negative_passages.docid: 86472
negative_passages.text: $f:\mathbb R \to \mathbb R^n$ be such that $G(f):=\{(x,f(x)):x \in \mathbb R\}$ is closed and connected in $\mathbb R^{n+1}$ , is $f$ 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: 134703
negative_passages.text: Let $X$ be a connected space, $f:X\to X$ continuous involution, $g:X\to\mathbb{R}$ continuous. Prove that $\exists$ $x\in X$ such that $g(x)=g(f(x))$.
negative_passages.docid: 62723
negative_passages.text: If $f\colon \mathbb{R} \to \mathbb{R}$ is such that $f (x + y) = f (x) f (y)$ and continuous at $0$, then continuous everywhere
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: 56413
negative_passages.text: $f(g(x))=g(f(x))$ implies $f(c)=g(c)$ for some $c$
negative_passages.docid: 12523
negative_passages.text: Supremum of a Continuous Function is Continuous
negative_passages.docid: 145919
negative_passages.text: If $f: \Bbb{R}\rightarrow\Bbb{R}$ is continuous at $0$ and $f(x)=f(2x)$ for each $x\in\Bbb{R}$ then $f$ is constant.
negative_passages.docid: 117502
negative_passages.text: Proving that a certain continuous function is surjective.
negative_passages.docid: 147299
negative_passages.text: Prove that $g(x,y) = \frac{x^3y}{x^2+y^2}$ with $g(0,0)=0$ is continuous
negative_passages.docid: 68048
negative_passages.text: Prove that if $f$ is differentiable on $[a,b]$ and $f'$ is monotonic on $[a,b]$ , $f'$ is continuous on $[a,b]$
negative_passages.docid: 150554
negative_passages.text: proof that $f'$ is continuous at $a$
negative_passages.docid: 71049
negative_passages.text: Functional equation $f(xy)=f(x)+f(y)$ and continuity
negative_passages.docid: 66391
negative_passages.text: Show that $f(x+y)=f(x)+f(y)$ implies $f$ continuous $\Leftrightarrow$ $f$ measurable