query: For given $a \in \mathbb{R}$ there exists unique continuous function $f:\mathbb{R} \to \mathbb{R}$.
positive_passages.docid: 11828
positive_passages.text: continuous functions on $\mathbb R$ such that $g(x+y)=g(x)g(y)$
negative_passages.docid: 53738
negative_passages.text: Monotone functions and continuity
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: 33308
negative_passages.text: Unique local extremum is absolute extremum for continuous functions
negative_passages.docid: 47
negative_passages.text: Let $f$ be a continuous and open map from $\mathbb R $ to $\mathbb R$.Prove that $f$ is monotonic.
negative_passages.docid: 124535
negative_passages.text: continuous function from $[\pi,2\pi]\to \mathbb{R}^2$
negative_passages.docid: 143174
negative_passages.text: property of continuous function
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: 91852
negative_passages.text: A continuous, injective function $f: \mathbb{R} \to \mathbb{R}$ is either strictly increasing or strictly decreasing.
negative_passages.docid: 37966
negative_passages.text: If $f$ is continuous and $f(x+y) = f(x)+f(y)$, then $f(x) = cx$ for all $x \in \mathbb{R}$
negative_passages.docid: 113503
negative_passages.text: Show that exists a not decreasing function that $f:(a,b)\rightarrow \mathbb{R}$ that is continuous only in $(a,b)\setminus D$.
negative_passages.docid: 9320
negative_passages.text: Extension of continuous function on a closed subset of $\mathbb{R}$
negative_passages.docid: 152784
negative_passages.text: $f \in {\mathscr R[a,b]} \implies f $ has infinitely many points of continuity.
negative_passages.docid: 113975
negative_passages.text: Let $f:\mathbb{R}\to\mathbb{R}$, $f(kx)=f(x)$. Show that if $f$ is continuous at $0$ then $f$ is a Constant function
negative_passages.docid: 139744
negative_passages.text: For a continuous function $f$ show $\exists c\in (0,1)$ s.t $f(c)=3c^2$
negative_passages.docid: 14495
negative_passages.text: I need to find all functions $f:\mathbb R \rightarrow \mathbb R$ which are continuous and satisfy $f(x+y)=f(x)+f(y)$
negative_passages.docid: 52922
negative_passages.text: let $f \colon [a,b] \to \mathbb{R}$ be continuous on $[a,b]$ and..