query: The set of functions $P$ for which $P(x)^2=P(x^2)$ for all $x\in \mathbb{R}$
positive_passages.docid: 141938
positive_passages.text: Find all continuous functions from positive reals to positive reals such that $f(x)^2=f(x^2)$
negative_passages.docid: 151947
negative_passages.text: Finding a pair of functions with properties
negative_passages.docid: 62239
negative_passages.text: For a polynomial $p(z)$, prove there exist $R>0$, such that if $|z|=R$, then $|p(z)|\geq |a_n|R^n/2$
negative_passages.docid: 63701
negative_passages.text: An example of a regular function over an open set
negative_passages.docid: 47992
negative_passages.text: Is there a function $f\colon\mathbb{R}\to\mathbb{R}$ such that every non-empty open interval is mapped onto $\mathbb{R}$?
negative_passages.docid: 39090
negative_passages.text: Finding functions such that $f(g(x)) = f(x)g(x)$
negative_passages.docid: 11828
negative_passages.text: continuous functions on $\mathbb R$ such that $g(x+y)=g(x)g(y)$
negative_passages.docid: 22464
negative_passages.text: Which of the following subsets of   $\mathbb{R}^2$ are compact
negative_passages.docid: 152451
negative_passages.text: REVISITED $^2$: Does a solution in $\mathbb{R}^n$ imply a solution in $\mathbb{Q}^n$?
negative_passages.docid: 2198
negative_passages.text: Maps from subsets of $\mathbb{R}^2$ that are either open/closed/continuous
negative_passages.docid: 36087
negative_passages.text: Every Function in a Finite Field is a Polynomial Function
negative_passages.docid: 62709
negative_passages.text: Functions that are sets of all function - proofs
negative_passages.docid: 23186
negative_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: 134043
negative_passages.text: Prove that for $ f(x+2\cdot\pi)=f(x), x \in \mathbb{R} $, there exists $ x_0 $ so $ f(x_0+\pi)=f(x_0) $
negative_passages.docid: 44704
negative_passages.text: familly of Radially open sets in $\mathbb{R}^2$
negative_passages.docid: 2527
negative_passages.text: Entire function bounded by a polynomial is a polynomial
negative_passages.docid: 90284
negative_passages.text: A function that is $L^p$ for all $p$ but is not $L^\infty$?