query: Question about a continuous function such that $f(x^2)=f(x)$.
positive_passages.docid: 58901
positive_passages.text: Why is every such function constant: $f(x)=f(x^2)$ for $x \in [0,1]$?
negative_passages.docid: 141938
negative_passages.text: Find all continuous functions from positive reals to positive reals such that $f(x)^2=f(x^2)$
negative_passages.docid: 150770
negative_passages.text: Why is $f^n(\xi(x))$ continuous over $x$?
negative_passages.docid: 126842
negative_passages.text: Fixed point in a continuous function
negative_passages.docid: 2321
negative_passages.text: Can there be two distinct, continuous functions that are equal at all rationals?
negative_passages.docid: 85092
negative_passages.text: Proving $ f(x) = x^2 $ is not uniformly continuous on the real line
negative_passages.docid: 72847
negative_passages.text: Does $f(x)$ is continuous and $f = 0$ a.e. imply $f=0$ everywhere?
negative_passages.docid: 68165
negative_passages.text: Find a function $f: \mathbb{R} \to \mathbb{R}$ that is continuous at precisely one point?
negative_passages.docid: 104044
negative_passages.text: Prove continuous functions $f$ and $g$ are equal
negative_passages.docid: 90565
negative_passages.text: If $f:[0,1] \rightarrow \mathbb{R}$ is continuous function such that $f(0)=f(1)$ then there exists $ x\in [0,1]$ such that $f(x) = f(x+\frac{1}{n})$.
negative_passages.docid: 120947
negative_passages.text: continuous real-valued function
negative_passages.docid: 97316
negative_passages.text: Let $F : X → X$ be continuous. Prove that the set $\{x ∈ X : F(x) = x\}$ of fixed points of F is closed in X
negative_passages.docid: 58664
negative_passages.text: Proving a function is continuous for a fixed variable
negative_passages.docid: 36983
negative_passages.text: Does there exist a continuous function such that f(x) is rational for every irrational x, and f(x) is irrational for every rational x?
negative_passages.docid: 128911
negative_passages.text: Continuous extension of a real function
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: 75735
negative_passages.text: Let $f$ be a continuous function on an interval around $0$ and let $a_i=f(\frac{1}{i})$ (for large enough $i$)