query: Prove or disprove that $f$ is identity function given that $f$ is continuous and $f(f(f(x)))=x$
positive_passages.docid: 16338
positive_passages.text: 3rd iterate of a continuous function equals identity function
negative_passages.docid: 113687
negative_passages.text: $f$ is continuous function from $ [0,1]$ to itself and f(0)=0, f(1)=1 and $f(f(x))=x$ then show that $f(x)=x$ on$[0,1]$
negative_passages.docid: 92376
negative_passages.text: Let $f:A \rightarrow B$ and C ⊂A. Define f[C]={b∈B: b=f(a) for some a∈C}. Prove or disprove each of the statement
negative_passages.docid: 80643
negative_passages.text: Show that f(x) defined on [0,infinity) R to is continuous at 0.
negative_passages.docid: 77074
negative_passages.text: If $ f(f(f(x)))=x$, does$ f(x)=x$ necessarily follow?
negative_passages.docid: 111756
negative_passages.text: How do I prove that then $f(x) = 1$ for some $x ∈ [0, 1]$?
negative_passages.docid: 55605
negative_passages.text: If $f_n(x_n) \to f(x)$ whenever $x_n \to x$, show that $f$ is continuous
negative_passages.docid: 131992
negative_passages.text: A problem on continuous functions
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: 94781
negative_passages.text: Prove that $M(t)=\sup_ {a \leq x \leq t} f(x)$ given $f(x)$ is continuous on $[a,b]$
negative_passages.docid: 71049
negative_passages.text: Functional equation $f(xy)=f(x)+f(y)$ and continuity
negative_passages.docid: 105700
negative_passages.text: Proving $\,f$ is constant.
negative_passages.docid: 132013
negative_passages.text: How do I show that if $f : [0,1] \to [0,1]$ is continuous, then there is some $c$ such that $f(c) = c$?
negative_passages.docid: 50012
negative_passages.text: $f: [a,b] \to \mathbb{R}$, $f(a) = f(b)$. Show $\exists c \in [a, \frac{a+b}{2} ]$ with $f(c)=f(c+\frac{b-a}{2})$
negative_passages.docid: 130023
negative_passages.text: Prove that the sum of two continuous functions is continuous
negative_passages.docid: 77267
negative_passages.text: Given $f$ is continuous and $f(x)=f(e^{t}x)$ for all $x\in\mathbb{R}$ and $t\ge0$, show that $f$ is constant function
negative_passages.docid: 72847
negative_passages.text: Does $f(x)$ is continuous and $f = 0$ a.e. imply $f=0$ everywhere?